IN   MEMORIAM 
FLORIAN  CAJORI 


THE 


LOGIC   AND   UTILITY 


OF 


MATHEMATICS, 


WITH  THE  BEST  METHODS   OF  INSTRUCTION"  EXPLAINED 
AND  ILLUSTRATED. 


BY  CHARLES  DAYIES,  LL.D. 


NEW  YORK: 
PUBLISHED  BY  A.  S.  BARNES  &  CO. 

NO.   51    JOHN-STEEET. 

CINCINNATI :— H.  W.  DERBY  &  COMPANY. 
1851. 


Entered  according  to  Act  of  Congress,  in  the  year  Eighteen  Hundred  and  fifty, 

BY  CHARLES  DAVIES, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern  District 
of  New  York. 


STEREOTYPED  BY 

BICHARD   C.  VALENTINE, 

NEW  YORK. 

F.  C.  GUTIERREZ,  Printer, 
No.  51  John-street,  corner  of  Dutch. 


PREFACE. 


THE  following  work  is  not  a  series  of  speculations.  It  is  but 
an  analysis  of  that  system  of  mathematical  instruction  which 
has  been  steadily  pursued  at  the  Military  Academy  over  a 
quarter  of  a  century,  and  which  has  given  to  that  institution 
its  celebrity  as  a  school  of  mathematical  science. 

It  is  of  the  essence  of  that  system  that  a  principle  be  taught 
before  it  is  applied  to  practice  ;  that  general  principles  and  gen 
eral  laws  be  taught,  for  their  contemplation  is  far  more  improving 
to  the  mind  than  the  examination  of  isolated  propositions  ;  and 
that  when  such  principles  and  such  laws  are  fully  compre 
hended,  their  applications  be  then  taught  as  consequences  or 
practical  results. 

This  view  of  education  led,  at  an  early  day,  to  the  union  of 
the  French  and  English  systems  of  mathematics.  By  this 
union  the  exact  and  beautiful  methods  of  generalization,  which 
distinguish  the  French  school,  were  blended  with  the  practical 
methods  of  the  English  system. 

The  fruits  of  this  new  system  of  instruction  have  been  abun 
dant.  The  graduates  of  the  Military  Academy  have  been 
sought  for  wherever  science  of  the  highest  grade  has  been 


PREFACE. 


needed.  Russia  has  sought  them  to  construct  her  railroads  ;* 
the  Coast  Survey  needed  their  aid ;  the  works  of  internal  im 
provement  of  the  first  class  in  our  country,  have  mostly  been 
conducted  under  their  direction ;  and  the  recent  war  with  Mexico 
afforded  ample  opportunity  for  showing  the  thousand  ways  in 
which  science — the  highest  class  of  knowledge — may  be  made 
available  in  practice. 

All  these  results  are  due  to  the  system  of  instruction.  In 
that  system  Mathematics  is  the  basis — Science  precedes  Art — 
Theory  goes  before  Practice — the  general  formula  embraces  all 
the  particulars. 

It  was  deemed  necessary  to  the  full  development  of  the  plan 
of  the  work,  to  give  a  general  view  of  the  subject  of  Logic. 
The  materials  of  Book  I.  have  been  drawn,  mainly,  from  the 
works  of  Archbishop  Whately  and  Mr.  Mill.  Although  the 
general  outline  of  the  subject  has  but  little  resemblance  to  the 
work  of  either  author,  yet  very  much  has  been  taken  from  both  ; 
and  in  all  cases  where  it  could  be  done  consistently  with  my  own 
plan,  I  have  adopted  their  exact  language.  This  remark  is  par 
ticularly  applicable  to  Chapter  III.,  Book  L,  which  is  taken, 
with  few  alterations,  from  Whately. 

For  a  full  account  of  the  objects  and  plan  of  the  work,  the 
reader  is  referred  to  the  Introduction. 

FISHKILL  LANDING, 
June,  1850. 


*  Major  Whistler,  the  engineer,  to  whom  was  intrusted  the  great  enterprise 
of  constructing  a  railroad  from  St.  Petersburg  to  Moscow,  and  Major  Brown, 
who  succeeded  him  at  his  death,  were  both  graduates  of  the  Military  Acad 
emy. 


CONTENTS 


INTRODUCTION. 

PAGE 

OBJECTS  AND  PLAN  OF  THE  WORK 11 


BOOK     I, 
LOGIC. 

CHAPTER  L 
DEFINITIONS — OPERATIONS  OF  THE  MIND — TERMS  DEFINED.  .   27 

SECTION 

Definitions 1—  6 

Operations  of  the  Mind  concerned  in  Reasoning 6 — 12 

Abstraction 12—14 

Generalization 14 — 22 

Terms — Singular  Terms — Common  Terms 15 

Classification 16 — 20 

Nature  of  Common  Terms 20 

Science 21 

Art  .  22 


CONTENTS. 


CHAPTER  II. 

PAGE 

SOURCES  AND  MEANS  OF  KNOWLEDGE — INDUCTION 41 

SECTION 

Knowledge 23 

Facts  and  Truths 24 — 27 

Intuitive  Truths 27 

Logical  Truths 28 

Logic 29 

Induction .  30—34 


CHAPTER  III. 

DEDUCTION — NATURE  OF  THE  SYLLOGISM — ITS  USES  AND  AP 
PLICATIONS  Page     54 

SECTION 

Deduction 34 

Propositions 35—40 

Syllogism 40 42 

Analytical  Outline  of  Deduction 42—67 

Aristotle's  Dictum 54 61 

Distribution  and  Non-distribution  of  Terms 61 — 67 

Rules  for  examining  Syllogisms 67 

Of  Fallacies 68 71 

Concluding  Remarks 71 75 


CONTENTS. 


BOOK    II, 
MATHEMATICAL   SCIENCE. 

CHAPTER  I. 

QUANTITY  AND  MATHEMATICAL  SCIENCE  DEFINED — DIFFER 
ENT  KINDS  OF  QUANTITY — LANGUAGE  OF  MATHEMATICS 
EXPLAINED — SUBJECTS  CLASSIFIED — UNIT  OF  MEASURE 
DEFINED — MATHEMATICS  A  DEDUCTIVE  SCIENCE  . . .  .Page  99 

SECTION 

Quantity 75 — 79 

Number '. 79—81 

Space 81—87 

Analysis 87 — 91 

Language  of  Mathematics 91 — 94 

Quantity  Measured 94—97 

Pure  Mathematics 97-101 

Comparison  of  Quantities 101 

Axioms  or  Formulas  for  inferring  Equality 102 

Axioms   or   Formulas  for  inferring  Inequality 102 

CHAPTER  II. 

PAGB 

ARITHMETIC — SCIENCE  AND  ART  OF  NUMBERS    117 

SECTION  I. 

SECTION 

First  Notions  of  Numbers  104—107 

Ideas  of  Numbers  Generalized 107 — 110 

Unity  and  a  Unit  Defined 110 

Simple  and  Denominate  Numbers Ill — 113 

Alphabet — Words — Grammar 113 

Arithmetical  Alphabet 114 

Spelling  and  Reading  in  Addition 115 — 120 

Spelling  and  Reading  in  Subtraction 120 — 122 


CONTENTS. 


SECTION 

Spelling  and  Reading  in  Multiplication 122 

Spelling  and  Reading  in  Division 123 

Units  increasing  by  the  Scale  of  Tens 124—131 

Units  increasing  by  Varying  Scales 131 

Integer  Units  of  Arithmetic 132 

Abstract  or  Simple  Units 132 — 134 

Units  of  Currency 134—136 

Units  of  Weight  136—139 

Units  of  Measure 139 — 150 

Advantages  of  the  System  of  Unities 150 

System  of  Unities  applied  to  the  Four  Ground  Rules.  151 — 155 

SECTION   II. 

Fractional  Units  changing  by  the  Scale  of  Tens 155—158 

Fractional  Units  in  general 158 — 161 

Advantages  of  the  System  of  Fractional  Units 161—163 

SECTION   III. 
Proportion  and  Ratio 163 — 172 

SECTION   IY. 
Applications  of  the  Science  of  Arithmetic 172 — 180 

SECTION   Y. 

Methods  of  teaching  Arithmetic  considered 180 

Order  of  the  Subjects 180 — 183 

1st.  Integer  Units 183—185 

2d.  Fractional  Units 185 

3d.  Comparison  of  Numbers,  or  Rule  of  Three 186—188 

4th.  Practical  Part,  or  Applications  of  Arithmetic 188 

Objections  to  Classification  answered  189 191 

Objections  to  the  new  Method 191 

Arithmetical  Language 192 200 

Necessity  of  exact  Definitions  and  Terms 200—206 

How  should  the  Subjects  be  presented 206—209 

Text-Books 209—214 

First  Arithmetic 214 227 


CONTENTS. 


SECTION 

Second  Arithmetic 227—231 

Third  Arithmetic 231—236 

Concluding  Remarks 236 

CHAPTER  III. 

GEOMETRY  DEFINED — THINGS  OF  WHICH  IT  TREATS — COM 
PARISON  AND  PROPERTIES  OF  FIGURES — DEMONSTRATION 
— PROPORTION — SUGGESTIONS  FOR  TEACHING  ....  Page  223 

SECTION 

Geometry 237 

Things  of  which  it  treats 238—248 

Comparison  of  Figures  with  Units  of  Measure 249 — 256 

Properties  of  Figures 256 

Marks  of  what  may  be  proved 257 

Demonstration 258—267 

Proportion  of  Figures 267—270 

Comparison  of  Figures 270 — 273 

Recapitulation — Suggestions  for  Teachers 273 

CHAPTER  IV. 

ANALYSIS — ALGEBRA — ANALYTICAL  GEOMETRY — DIFFEREN 
TIAL  AND  INTEGRAL  CALCULUS Page  261 

SECTION 

Analysis 274—280 

Algebra 280 

Analytical  Geometry 281—283 

Differential  and  Integral  Calculus 283 — 286 

Algebra  further  considered 286—296 

Minus  Sign 296—298 

Subtraction 298 

Multiplication 299—302 

Zero  and  Infinity 302—307 

Of  the  Equation 307— 31 1 

Axioms 311 

Equality — its  meaning  in  Geometry 312 

Suggestions  for  those  who  teach  Algebra 315 


L 


10  CONTENTS. 


BOOK   III, 

UTILITY  OF   MATHEMATICS. 

CHAPTER  I. 

THE  UTILITY  OF  MATHEMATICS  CONSIDERED  AS  A  MEANS  OF 
INTELLECTUAL  TRAINING  AND  CULTURE  .  .  Page  293 

O 

CHAPTER  II. 

THE  UTILITY  OF  MATHEMATICS  REGARDED  AS  A  MEANS  OF 
ACQUIRING  KNOWLEDGE — BACONIAN  PHILOSOPHY 308 

CHAPTER  III. 

THE  UTILITY  OF  MATHEMATICS  CONSIDERED  AS  FURNISHING 
THOSE  RULES  OF  ART  WHICH  MAKE  KNOWLEDGE  PRACTI 
CALLY  EFFECTIVE  .  .  325 


APPENDIX. 

A  COURSE  OF  MATHEMATICS — WHAT  IT  SHOULD  BE 341 

ALPHABETICAL  INDEX   353 


INTRODUCTION 


OBJECTS   AND  PLAN  OF  THE  WORK. 

UTILITY   and   Progress    are  the  two   leading     utility 
ideas  of  the  present  age.     They  were  manifested    Progress: 
in  the  formation  of  our  political  and  social  insti-   Their  influ- 

Gncc  in  ffov* 

tutions,  and  have  been  further  developed  in  the    eminent: 
extension  of  those  institutions,  with  their  subdu 
ing  and  civilizing  influences,  over  the  fairest  por 
tions  of  a  great  continent.     They  are  now  be 
coming  the  controlling  elements  in  our  systems  in  education, 
of  public  instruction. 

What,  then,  must  be  the  basis  of  that  system      what 

the  basis  of 

of  education  which  shall  embrace  within  its  ho-  utility  and 
rizon  a  Utility  as  comprehensive  and  a  Progress 
as  permanent  as  the  ordinations  of  Providence, 
exhibited  in  the  laws  of  nature,  as  made  known 
'by  science?  It  must  obviously  be  laid  in  the 
examination  and  analysis  of  those  laws ;  and 


12  INTRODUCTION. 


.  primarily;  in  those  preparatory  studies  which  fit 
^?';  :  %  arid  .  qualify  trie  mind  for  such  Divine  Contem 
plations. 

Bacon's         When  Bacon  had  analyzed  the  philosophy  of 

Philosophy. 

the  ancients,  he  found  it  speculative.  The  great 
highways  of  life  had  been  deserted.  Nature, 
spread  out  to  the  intelligence  of  man,  in  all  the 
minuteness  and  generality  of  its  laws — in  all  the 
harmony  and  beauty  which  those  laws  develop — 
had  scarcely  been  consulted  by  the  ancient  phi- 
Phiioso-  losophers.  They  had  looked  within,  and  not 

phy  of  the 

Ancients,  without.  They  sought  to  rear  systems  on  the 
uncertain  foundations  of  human  hypothesis  and 
speculation,  instead  of  resting  them  on  the  im 
mutable  laws  of  Providence,  as  manifested  in 
the  material  world.  Bacon  broke  the  bars  of 
this  mental  prison-house :  bade  the  mind  go  free, 
and  investigate  nature. 

Foundations       Bacon  laid  the  foundations  of  his  philosophy  in 
Phiklophy :  orgarnc  laws>  and  explained  the  several  processes 
of  experience,  observation,  experiment,  and  in 
duction,  by  which  these  laws  are  made  known, 
why  op-     He  rejected  the  reasonings  of  Aristotle  because 

posed  to  Aris 
totle's,      they  were   not  progressive   and  useful ;  because 

they  added  little  to  knowledge,  and  contributed 
nothing  to  ameliorate  the  sufferings  and  elevate 
the  condition  of  humanity. 


PLAN     OF     THE     WORK.  13 

The  time  seems  now  to  be  at  hand  when  the     Practical 
philosophy  of  Bacon  is  to  find  its  full  develop 
ment.     The  only  fear  is,  that  in  passing  from  a 
speculative  to  a  practical  philosophy,  we  may, 
for  a  time,  lose  sight  of  the  fact,  that  Practice 
without  Science  is   Empiricism;    and   that    all   its  true  na- 
which  is  truly  great  in  the  practical  must  be  the 
application  and  result  of  an  antecedent  ideal. 


What,  then,  are  the  sources  of  that  Utility,     what  is 
and  the  basis  of  that  Practical,  which  the  pres-  temofedu- 
ent  generation  desire,  and  after  which  they  are 
so  anxiously  seeking  ?     What  system  of  training 
and  discipline  will  best  develop  and  steady  the 
intellect  of  the  young  ;  give  vigor  and  expan 
sion  to  thought,  and  stability  to  action  ?     What  which  win 
course  of  study  will  most  enlarge  the  sphere  of   steady  the 
investigation  ;  give  the  greatest  freedom  to  the 
mind  without    licentiousness,    and   the   greatest 
freedom  to  action  consistent  with  the  laws  of 
nature,  and  the  obligations  of  the  social  com 
pact  ?     What  subject  of  study  is,  from  its  na-    what  are 

.    .  the  subjects 

ture,   most   likely  to   ensure   this   training,   and 


contribute  to  such  results,  and  at  the  same  time 

lay  the  foundations  of  all  that  is  truly  great  in 

the  Practical  ?     It  has  seemed  to  me  that  math-  Mathematics. 

ematical  science  may  lay  claim  to  this  pre-emi 

nence. 


14  INTRODUCTION. 


The  first  impressions  which  the  child  receives 
of  Number  and  Quantity  are  the  foundations  of 
knowledge,  j^  mathematical  knowledge.     They  form,  as  it 
were,  a  part  of  his  intellectual  being.     The  laws 
Laws  of     of  Nature  are  merely  truths  or  generalized  facts, 

Nature.  .  . 

in  regard  to  matter,  derived  by  induction  from 

experience,  observation,  and  experiment.      The 

laws  of  mathematical    science   are   generalized 

Number     truths  derived  from  the  consideration  of  Number 

ga^       and  Space.      All  the  processes  of  inquiry  and 

investigation  are  conducted   according  to  fixed 

laws,  and  form  a  science  ;  and  every  new  thought 

and  higher  impression  form  additional  links  in 

the  lengthening  chain. 

Mathemat-        The  knowledge  which  mathematical  science 

ical  knowl-     . 

edge .      imparts  to  the  mind  is  deep — profound — abiding. 

It  gives  rise  to  trains  of  thought,  which  are  born 

in  the  pure  ideal,  and  fed  and  nurtured  by  an 

acquaintance  with  physical  nature  in  all  its  mi- 

what  it    nuteness  and  in  all  its  grandeur :  which  survey 

does.  the  laws  of  elementary  organization,  by  the  mi 
croscope,  and  weigh  the  spheres  in  the  balance 
of  universal  gravitation. 

What  The  processes  of  mathematical  science  serve 
to  give  mental  unity  and  wholeness.  They  im 
part  that  knowledge  which  applies  the  means  of 


PLAN     OF     THE     WORK.  15 

crystallization  to  a  chaos  of  scattered  particulars,  Right  knowi- 
and  discovers  at  once  the  general  law,  if  there  tL^eanlTof 
be  one,  which  forms  a  connecting  link  between    cryt^liza" 
them.      Such   results   can  only  be  attained  by 
minds  highly  disciplined  by  scientific  combina 
tions.     In  all  these  processes  no  fact  of  science 
is  forgotten  or  lost.     They  are  all  engraved  on 
the  great  tablet  of  universal  truth,  there  to  be 
read  by  succeeding  generations  so  long  as  the    it  records 

.  .,        .      .  .     and  preserves 

laws  of  mind  remain  unchanged.     This  is  stri-      truth. 
kingly  illustrated  by  the  fact,  that  any  diligent 
student  of  a  college  may  now  read  the  works  of 
Newton,  or  the  Mecanique  Celeste  of  La  Place. 

The  educator  regards  mathematical  science     HOW  the 

educator  re- 

as  the  great  means  of  accomplishing  his  work,  gardsmatn- 
The  definitions  present  clear  and  separate  ideas, 
which  the  mind  readily  apprehends.    The  axioms  The  axioms. 
are  the  simplest  exercises  of  the  reasoning  fac 
ulty,  and  afford  the  most  satisfactory  results  in 
the  early  use  and  employment  of  that  faculty. 
The  trains  of  reasoning  which  follow  are  com 
binations,   according   to   logical    rules,  of  what 
has  been  previously   fully   comprehended;    and  influence  of 

,  the  study  of 

the  mind  and  the   argument  grow  together,  so  mathematica 
that  the  thread  of  science  and  the  warp  of  the  onthemind- 
intellect  entwine  themselves,  and  become  insep 
arable.     Such  a  training  will  lay  the  foundations 


16  INTRODUCTION. 

of  systematic  knowledge,  so  greatly  preferable 
to  conjectural  judgments. 

HOW  the         The  philosopher  regards  mathematical  science 

P  regTrds**    as  tne  mere  tools  of  his   higher  Vocation.       Look- 
mathematics:  ^  w-t^   a  stea(jy  anc[  anxious   eye   to  Nature, 

and  the  great  laws  which  regulate  and  govern 
all  things,  he  becomes  earnestly  intent  on  their 
examination,  and  absorbed  in  the  wonderful  har 
monies  which  he  discovers.  Urged  forward  by 
its  necessity  these  high  impulses,  he  sometimes  neglects  that 

to  him. 

thorough  preparation,  in  mathematical  science, 
necessary  to  success;  and  is  not  unfrequently 
obliged,  like  Antaeus,  to  touch  again  his  mother 
earth,  in  order  to  renew  his  strength. 


The  views        The  mere  practical  man  regards  with   favor 

of  the  practi 
cal  man.     only  the  results  of  science,  deeming  the  reason 
ings  through  which  these  results  are  arrived  at, 
quite  superfluous.     Such  should  remember  that 

instruments  the  mind  requires   instruments    as  well    as    the 

of  the  mind. 

hands,  and  that  it  should  be  equally  trained  in 
their  combinations  and  uses.  Such  is,  indeed, 
now  the  complication  of  human  affairs,  that  to 
do  one  thing  well,  it  is  necessary  to  know  the 
properties  and  relations  of  many  things.  Every 
Everything  thing,  whether  existing  in  the  abstract  or  in  the 

has  a  law. 

material  world ;  whether  an  element  of  knowl- 


PL  AN     OF     THE     WORK.  17 


edge  or  a  rule  of  art,  has  its  connections  and  its     TO  know 

...          the  law  is  to 

law:  to  understand  these  connections  and  that    knowthe 
law,  is  to  know  the  thing.     When  the  principle 
is  clearly  apprehended,  the  practice  is  easy. 


With  these  general  views,  and  under  a  firm  Mathematics 
conviction  that  mathematical  science  must  be 
come  the  great  basis  of  education,  I  have  be 
stowed  much  time  and  labor  on  its  analysis,  as 
a  subject  of  knowledge.  I  have  endeavored  to 
present  its  elements  separately,  and  in  their  con-  HOW. 
nections ;  to  point  out  and  note  the  mental  fac 
ulties  which  it  calls  into  exercise  ;  to  show  why 
and  how  it  develops  those  faculties;  and  in  what 
respect  it  gives  to  the  whole  mental  machinery 
greater  power  and  certainty  of  action  than  can 
be  attained  by  other  studies.  To  accomplish  what  was 

deemed  ne- 

these  ends,  in  the  way  that  seemed  to  me  most     cessary. 
desirable,   I  have  divided  the  work   into   three 
parts,  arranged  under  the  heads  of  Book  I.,  II., 
and  III. 


Book  I.  treats  of  Logic,  both  as  a  science  and      Logic, 
an  art ;  that  is,  it  explains  the  laws  which  gov 
ern    the  reasoning   faculty,   in    the    complicated 
processes  of  argumentation,  and  lays  down  the  Explanation, 
rules,  deduced  from   those  laws,  for  conducting    • 
such   processes.      It   being  one   of  the   leading 

2 


18  INTRODUCTION. 


For  what    objects  to  show  that  mathematical  science  is  the 

sed'       best  subject  for  the  development  and  application 

of  the  principles  of  logic  ;  and,  indeed,  that  the 

science  itself  is  but  the  application  of  those  prin- 

The  necessity  ciples   to  the    abstract    quantities  Number    and 

of  treating  it.  .  I 

Space,  it  appeared  indispensable  to  give,  in  a 
manner  best  adapted  to  my  purpose,  an  out 
line  of  the  nature  of  that  reasoning  by  means 
of  which  all  inferred  knowledge  is  acquired. 

Book  ii.          Book   II.    treats    of    Mathematical    Science. 

Here  I  have  endeavored  to  explain  the  nature  of 

or  what  it    the  subjects  with  which  mathematical  science  is 

treats.  1-1  i  •    i          • 

conversant ;  the  ideas  which  arise  in  examining 
and  contemplating  those  subjects ;  the  language 
employed  to  express  those  ideas,  and  the  laws  of 
their  connection.  This,  of  course,  led  to  a  class- 
Manner  of  ification  of  the  subjects;  to  an  analysis  of  the 

treating. 

language  used,  and  an  examination  of  the  reason 
ings  employed  in  the  methods  of  proof. 


Book  m.        Book  III.  explains  and  illustrates  the  Utility  of 

Utility  of      ,,  -TV 

Mathematics.  Mathematics  :  First,  as  a  means  of  mental  disci 
pline  and  training ;  Secondly,  as  a  means  of  ac 
quiring  knowledge ;  and,  Thirdly,  as  furnishing 
those  rules  of  art,  which  make  knowledge  prac- 
acally  effective. 


PLANOFTHEWORK.  19 


Having  thus  given  the  general  outlines  of  the    classes  of 
work,  we  will  refer  to  the  classes  of  readers  for 
whose  use  it  is  designed,  and  the  particular  ad 
vantages  and  benefits  which  each  class  may  re 
ceive  from  its  perusal  and  study. 

There  are  four  classes  of  readers,  who  may,  Four  classes 
it  is   supposed,  be  profited,  more  or  less,  by  the 
perusal  of  this  work  : 

1st.  The  general  reader  ;  First  class. 

2d.  Professional  men  and  students  ;  second. 

3d.  Students  of  mathematics  and  philosophy  ;        Third. 

4th.  Professional  Teachers.  Fourth. 

First.  The  general  reader,  who  reads  for  im.   Advantages 
provement,  and  desires  to    acquire   knowledge,   eraireader. 
must  carefully  search  out  the  import  of  language. 
He  must  early  establish  and  carefully  cultivate 
the  habit  of  noting  the  connection  between  ideas     connec- 
and  their  signs,  and  also  the  relation  of  ideas  to  ^ordsTud" 
each   other.     Such   analysis   leads   to  attentive 
reading,  to  clear    apprehension,  deep  reflection, 
and  soon  to  generalization. 

Logic  considers  the  forms  in  which  truth  must      Logic, 
be  expressed,  and  lays  down  rules  for  reducing 
all  trains  of  thought  to  such  known  forms.     This 
habit  of  analyzing  arms  us  with  tests  by  which    its  value: 
we  separate  argument  from  sophistry — truth  from 
falsehood.     The   application  of  these  principles, 


INTRODUCTION. 


in  the  study  in  the  construction  of  the  mathematical  science, 
mathematics  where  the  relation  between  the  sign  (or  language) 
and  the  thing  signified  (or  idea  expressed),  is  un 
mistakable,  gives  precision  and  accuracy,  leads 
to  right  arrangement  and  classification,  and  thus 
prepares  the  mind  for  the  reception  of  general 
knowledge. 


Advantages       Secondly.    The  increase  of  knowledge  carries 

professio 
a)  men. 


with  it  the  necessity  of  classification.     A  limited 


number  of  isolated  facts  may  be  remembered,  or 
a  few  simple  principles  applied,  without  tracing 
out  their  connections,  or  determining  the  places 
which  they  occupy  in  the  science  of  general 
knowledge.  But  when  these  facts  and  principles 
are  greatly  multiplied,  as  they  are  in  the  learned 
The  reason,  professions  ;  when  the  labors  of  preceding  gen 
erations  are  to  be  examined,  analyzed,  compared ; 
when  new  systems  are  to  be  formed,  combining 
all  that  is  valuable  in  the  past  with  the  stimu 
lating  elements  of  the  present,  there  is  occasion 
for  the  constant  exercise  of  our  highest  facul- 
ties.  Knowledge  reduced  to  order  ;  that  is, 
knowledge  so  classified  and  arranged  as  to  be 
easi}y  remembered,  readily  referred  to,  and  ad 
vantageously  applied,  will  alone  suffice  to  sift 
the  pure  metal  from  the  dust  of  ages,  and  fashion 
it  for  present  use.  Such  knowledge  is  Science. 


PLAN     OF     THE     WORK. 


Masses  of  facts,  like  masses  of  matter,  are  ca-    Knowledge 
pable  of  very  minute  subdivisions  ;  and  when  we  duced  to  te 
know  the  law  of  combination,  they  are  readily    elements- 
divided  or  reunited.     To  know  the  law,  in  any 
case,  is  to  ascend  to  the  source  ;   and  without 
that  knowledge  the  mind  gropes  in  darkness. 

It  has  been  my  aim  to  present  such  a  view    objects  of 
of  Logic    and  Mathematical  Science  as  would 
clearly  indicate,  to  the  professional  student,  and 
even  to  the  general  reader,  the  outlines  of  these 
subjects.      Logic  exhibits    the  general    formula    Logic  and 

,.       ,  ,  11     i  •      i  ^  i    mathematics. 

applicable  to  all  kinds  of  argumentation,  and 
mathematics  is  an  application  of  logic  to  the 
abstract  quantities  Number  and  Space. 

When  the  professional  student  shall  have  ex 
amined  the  subject,  even  to  the  extent  to  which  certainty  of 
it  is  here  treated,  he  will  be  impressed  with  the 
clearness,  simplicity,  certainty,  and  generality  of 
its  principles ;  and  will  find  no  difficulty  in  ma 
king  them  available  in  classifying  the  facts,  and 
examining  the  organic  laws  wrhich  characterize 
his  particular  department  of  knowledge. 

Thirdly.  Mathematical  knowledge  differs  from  Mathemati- 
every  other  kind  of  knowledge  in  this  :  it  is,  as   ca  edn°w 
it  were,  a  web  of  connected  principles  spun  out 
from  a  few  abstract  ideas,  until  it  has  become 
one  of  the  great  means   of  intellectual  develop-    its  extent 


22  INTRODUCTION. 


ment  and  of  practical  utility.     And  if  I  am  per- 

Necessity     mitted  to  extend  the  figure,  I  may  add,  that  the 

at  the  right  web  of  the  spider,  though  perfectly  simple,  if  we 

see  the  end  and  understand  the  way  in  which 

it  is  put  together,  is  yet  too  complicated  to  be 

unravelled,  unless  we  begin  at  the  right  point, 

and  observe  the  law  of  its  formation.     So  with 

mathematical  science.     It  is  evolved  from  a  few 

— a  very  few — elementary  and  intuitive  princi- 

HOW       pies  :  the  law  of  its  evolution  is  simple  but  ex- 

mathemati- 

cai  science  is  acting,  and  to  begin  at  the  right  place  and  pro 
ceed  in  the  right  way,  is  all  that  is  necessary  to 
make  the  subject  easy,  interesting,  and  useful, 
what  has        I  have  endeavored  to  point  out  the  place  of 

been  at 
tempted,     beginning,  and  to  indicate  the  way  to  the  math 
ematical  student.     I  am  aware  that  he  is  start 
ing  on  a  road  where  the  guide-boards  resemble 
each  other,   and  where,  for  the  want  of  careful 
observation,  they   are  often  mistaken ;    I    have 
sought,  therefore,  to  furnish  him  with  the  maps 
and  guide-books  of  an  old  traveller. 
Advantages       By  explaining  with   minuteness   the   subjects 

of  examining  . 

the  whole  about  which  mathematical  science  is  conversant, 
the  whole  field  to  be  gone  over  is  at  once  sur 
veyed:  by  calling  attention  to  the  faculties  of 

of^ntdTr!  the  m'md  which  the  science  brings  into  exercise, 
ing  the  men- we  are  better  prepared  to  note  the  intellectual 

tal  faculties : 

operations  which  the  processes  require  ;  and  by 


PLANOFTHEWORK.  23 


a  knowledge  of  the  laws  of  reasoning,  and  an  ofaknowi- 

.,  edge  of  the 

acquaintance  with  the  tests  of  truth,  we  are  en-  iawsofrea. 
abled  to  verify  all  our  results.    These  means  have 
been   furnished  in   the   following  work,   and  to 
aid  the  student  in  classification  and  arrangement, 
diagrams  have  been  prepared  exhibiting  separ-    what  has 

.  been  done. 

ately  and  m  connection  all  the  principal  parts  ot 
mathematical  science.  The  student,  therefore, 
who  adopts  the  system  here  indicated,  will  find 
his  way  clearly  marked  out,  and  will  recognise,  Advanta^e3 

J  J  to  the  stu- 

from  their  general  resemblance  to  the  descrip-  <ient. 
tions,  all  the  guide-posts  which  he  meets.  He 
will  be  at  no  loss  to  discover  the  connection 
between  the  parts  of  his  subject.  Beginning 
with  first  principles  and  elementary  combina 
tions,  and  guided  by  simple  laws,  he  will  go  for-  Where 

he  begins. 

ward  from  the  exercises  of  Mental  Arithmetic 
to  the  higher  analysis  of  Mathematical  Science 
on  an  ascent  so  gentle,  and  with  a  progress  so  °rder 

of  progress. 

steady,  as  scarcely  to  note  the  changes.  And 
indeed,  why  should  he  ?  For  all  mathematical 
processes  are  alike  in  their  nature,  governed  by 
the  same  laws,  exercising  the  same  faculties,  unity  of 

,,.,,.  .      ,  .  the  subject. 

and  lifting  the  mind  towards  the  same  eminence. 


Fourthly.  The  leading  idea,  in  the  construe-  Advantages 
tion  of  the  work,  has  been,  to  afford  substantial  professional 
aid  to  the  professional  teacher.  The  nature  of  teacher- 


24  INTRODUCTION. 


His  duties:  his  duties  —  their   inherent  difficulties  —  the  per- 
plexities  which  meet  him  at  every  step  —  the  want 


ments  and  .       ,  .     ,  (.     ,. 

difficulties:  oi  sympathy  and  support  in  his  hours  01  discour 
agement  —  (and  they  are  many)  —  are  circum 
stances  which  awaken  a  lively  interest  in  the 
hearts  of  all  who  have  shared  the  toils,  and  been 
themselves  laborers  in  the  same  vineyard.  He 
takes  his  place  in  the  schoolhouse  by  the  road 
side,  and  there,  removed  from  the  highways  of 

Remoteness  life,  spends  his  days  in  raising  the  feeble  mind 

from  active        ,.      ,  . 

life.  or  childhood  to  strength  —  in  planting  aright  the 
seeds  of  knowledge  —  in  curbing  the  turbulence 
of  passion  —  in  eradicating  evil  and  inspiring 
good.  The  fruits  of  his  labors  are  seen  but 
once  in  a  generation.  The  boy  must  grow  to 
Fruits  of  manhood  and  the  girl  become  a  matron  before 

his  efforts, 

he  is  certain  that  his  labors  have  not  been  in 


vain. 

Yet,  to  the  teacher  is  committed  the  high  trust 

of  forming  the  intellectual,  and,  to  a  certain  ex 

tent,   the  moral  development  of  a  people.     He 

Theimpor-   holds  in  his  hands   the  keys  of  knowledge      If 

tance  of  his     ,         „ 

labors.  tne  nrst  moral  impressions  do  not  spring  into 
life  at  his  bidding,  he  is  at  the  source  of  the 
stream,  and  gives  direction  to  the  current.  Al 
though  himself  imprisoned  in  the  schoolhouse, 
his  influence  and  his  teachings  affect  all  condi 
tions  of  society,  and  reach  over  the  whole  hori- 


PLAN     OF     THE     WORK. 


zon    of   civilization.      He  impresses   himself  on  The  influence 

r       ,  .  i   •    i       i          T  i    °f  h's  labors. 

the  young  of   the  age  in   which   he    lives,   and 
lives  again  in  the  age  which  succeeds  him. 


All   good  teaching  must   flow  from   copious    sources  of 

good  teach- 

knowledge.     The  shallow  fountain  cannot  emit        ing. 
a  vigorous  stream.     In  the  hope  of  doing  some 
thing    that   may   be    useful    to    the    professional 
teacher,   I    have    attempted    a    careful    and   full    ObJectsfor 

which  the 

analysis  of  mathematical  science.     I  have  spread    work  was 

undertaken. 

out,  in  detail,  those  methods  which  nave   been 

carefully  examined  and  subjected  to  the  test  of 

long  experience.      If  they  are  the  right  meth-    principles 

ods,  they  will  serve   as   standards  of  teaching ; 

for,  the  principles  of  imparting  instruction  are 

the  same  for  all  branches  of  knowledge. 


The  system  which  I  have  indicated  is  com 
plete  in  itself.  It  lays  open  to  the  teacher  the 
entire  skeleton  of  the  science — exhibits  all  its 
parts  separately  and  in  their  connection.  It 
explains  a  course  of  reasoning  simple  in  itself, 
and  applicable  not  only  to  every  process  in 
mathematical  science,  but  to  all  processes  of 
argumentation  in  every  subject  of  knowledge. 

The  teacher  who  thus  combines  science  with 
art,  no  longer  regards  Arithmetic  as  a  mere 
treadmill  of  mechanical  labor,  but  as  a  means — 


System. 


What  it 
presents. 


What  it 
explains. 


Science 
combined 
with  art: 


26  INTRODUCTION. 


and  the  simplest  means — of  teaching  the  art  and 

tages  result-         .  -  .  .  ,       ,  . 

ing  from  it.    science  of  reasoning  on   quantity — and  this   is 

the  logic   of  mathematics.     If  he  would  accom- 

Resuitsof    plish   well   his   work,    he  must   so   instruct  his 

right  instruc 
tion,       pupils  that   they  shall  apprehend  clearly,  think 

quickly  and  correctly,  reason  justly,  and  open 
their  minds  freely  to  the  reception  of  all  knowl 
edge. 


BOOK      I. 

LOGIC, 


CHAPTER    I. 

DEFINITIONS  -  OPERATIONS  OF  THE  MIND  -  TERMS  DEFINED. 
DEFINITIONS. 

§  1.  DEFINITION  is  a  metaphorical  word,  which    Definition 
literally    signifies    "  laying   down    a   boundary."  metaphorical 
All  definitions  are  of  names,  and  of  names  only  ; 

Some 

but  in  some  definitions,  it  is   clearly  apparent,    definitions 
that  nothing  is  intended  except  to  explain  the       ^ 
meaning  of  the  word  ;  while  in  others,  besides 

,          others  imply 

explaining  the  meaning  of  the  word,  it  is  also      tllings 


implied  that  there  exists,  or  may  exist,  a  thing 
corresponding  to  the  word.  words- 


§  2.  Definitions  which  do  not  imply  the  exist-  or  definitions 

...  which  do 

ence  of  things  corresponding  to   the  words  de-     not  imply 


fined,  are  those  usually  found  in  the  Dictionary 

of  one's  own  language.     They  explain  only  the     to  words. 


28  LOGIC.  [BOOK  i. 

Ther       meaning  of  the  word  or  term,  by  giving  some 
explain     equivalent  expression  which  may  happen  to  be 

words  by 

equivalents,  better  known.  Definitions  which  imply  the  ex 
istence  of  things  corresponding  to  true  words  de 
fined,  do  more  than  this. 

Definition         pOr  example :  "  A  triangle  is  a  rectilineal  fig- 

of  a 

triangle;     ure  having  three  sides."      This   definition    does 
w!tiat       two  things : 

implies.          i$t    it  explains  the  meaning  of  the  word  tri 
angle;  and, 

2d.  It  implies  that  there  exists,  or  may  exist, 
a  rectilineal  figure  having  three  sides. 


ofa  §  3.  To  define  a  word  when  the  definition  is 

definition 

which  im-   to   imply  the  existence   of  a  thing,   is   to  select, 

plies  the  ex-   /»  u      i  •  r     i          i  • 

isuaice  of    "om  a^  the  properties  of  the  thing  those  which 
a  thmg.      are  most  simp]6j  genera^  anc[  obvious  ;  and  the 


Properties    properties  must  be  very  well  known  to  us  before 

must  be  . 

known.      we  can  decide  which  are  the  fittest  for  this  pur 

pose.     Hence,  a  thing  may  have  many  properties 

besides  those  which  are  named  in  the  definition 

A  definition  of  the  word  which  stands  for  it.     This  second 

supports          . 

truth.  kind  of  definition  is  not  only  the  best  form  of  ex 
pressing  certain  conceptions,  but  also  contributes 
to  the  development  and  support  of  new  truths. 


In  §  4.  In  Mathematics,  and  indeed,  in  all  strict 

Mathematics 

names  imply  sciences,  names  imply  the  existence  of  the  things 


CHAP.   I.] 


DEFINITIONS. 


29 


things 

and 

express 

attributes. 


which  they  name ;  and  the  definitions  of  those 
names  express  attributes  of  the  things ;  so  that 
no  correct  definition  whatever,   of  any  mathe 
matical   term,   can  be  devised,  which  shall  not 
express  certain  attributes  of  the  thing  correspond 
ing  to  the  name.     Every  definition  of  this  class    Definition3 
is  a  tacit  assumption  of  some  proposition  which  ofthisclas3 
is    expressed    by    means    of   the    definition,  and  propositions, 
which  gives  to  such  definition  its  importance. 


§  5.  All  the  reasonings  in  mathematics,  which  Reasoning 

rest    ultimately  on    definitions,   do,  in  fact,  rest  restingon 

J  definitions ; 

on   the    intuitive    inference,    that    things    corre- 


rests  on 


spending  to  the  words  defined  have   a  conceiv-     intuitive 
able  existence  as  subjects  of  thought,  and  do  or 
may  have  proximately,  an  actual  existence.* 


*  There  are  four  rules  which  aid  us  in  framing  defini 
tions. 

1st.  The  definition  must  be  adequate :  that  is,  neither  too 
extended,  nor  too  narrow  for  the  word  defined. 

2d.  The  definition  must  be  in  itself  plainer  than  the  word 
defined,  else  it  would  not  explain  it. 

3d.  The  definition  should  be  expressed  in  a  convenient 
number  of  appropriate  words. 

4th.  When  the  definition  implies  the  existence  of  a  thing 
corresponding  to  the  word  defined,  the  certainty  of  that 
existence  must  be  intuitive. 


Four  rules. 


1st  rule. 


2cl  rule. 


3d  rule. 


4th  rule. 


30 


LOGIC.  [BOOK  i. 


OPERATIONS  OF  THE  MIND  CONCERNED  IN  REASONING. 

Three  opera-      §  g.  There  are  three  operations  of  the  mind 
tions  of  the  which  are  immediately  concerned  in  reasoning. 

1st.    Simple    apprehension  ;    2d.    Judgment ; 
3d:  Reasoning  or  Discourse. 

dim  lea          $  7-   Simple   apprehension   is   the   notion    (or 

prehension,  conception)  of  an  object  in  the  mind,  analogous 
to  the  perception  of  the  senses.  It  is  either 

incompiex.  Incomplex  or  Complex.  Incomplex  Apprehen 
sion  is  of  one  object,  or  of  several  without  any 
relation  being  perceived  between  them,  as  of  a 

Complex,  triangle,  a  square,  or  a  circle :  Complex  is  of 
several  with  such  a  relation,  as  of  a  triangle 
within  a  circle,  or  a  circle  within  a  square. 

§  8.  Judgment  is  the  comparing  together  in 
the  mind  two  of  the  notions  (or  ideas)  which 

Judgment 

defined,     are  the  objects  of  apprehension,  whether  com 
plex  or  incompiex,   and  pronouncing  that  they 
agree  or  disagree  with  each  other,  or  that  one 
of  them  belongs  or  does  not  belong  to,  the  other : 
for  example :  that  a  right-angled  triangle  and  an 
Judgment    equilateral  triangle  belong  to  the  class  of  figures 
either      called  triangles  ;  or  that  a  square  is  not  a  circle. 
or  l ''    Judgment,  therefore,  is  either  Affirmative  or  Neg- 
ncgative-     ative. 


CHAP.  I.]  ABSTRACTION.  31 

§  9.    Reasoning   (or  discourse)   is  the  act  of  reasoning 
proceeding  from  certain  judgments   to  another 
founded  upon  them  (or  the  result  of  them). 

§  10.    Language  affords  the  signs  by  which  Language 

these  operations  of  the  mind  are  recorded,  ex-  8ignsof 

pressed,   and  communicated.     It  is  also  an  in-  thousnt: 

strument  of  thought,   and  one  of  the  principal  also,  an 

instrument 

helps  in  all  mental  operations;  and  any  imper-   of  thought. 
fection   in    the   instrument,   or  in  the   mode  of 
using  it,  will  materially  affect  any  result  attained 
through  its  aid. 


§  11.  Every  branch  of  knowledge  has,   to  a 

Every  branch 

certain   extent,   its   own   appropriate   language ;  ofknowiedge 

has  its  own 

and  for  a   mind    not  previously  versed  in   the    language, 
meaning  and  right  use  of  the  various  words  and 

which 

signs  which  constitute  the  language,  to  attempt     must  be 

learned. 

the  study  of  methods  of  philosophizing,  would 
be  as  absurd  as  to  attempt  reading  before  learn 
ing  the  alphabet. 


ABSTRACTION. 

§  12.  The  faculty  of  abstraction  is  that  power 

f.     ,  •     i       ,  .  Abstraction, 

ol  the  mind  which  enables  us,  in  contemplating 
any  object  (or  objects),  to  attend  exclusively  to 


32  LOGIC.  [BOOK  i. 


some  particular  circumstance  belonging  to  it,  and 
quite  withhold  our  attention  from  the  rest.    Thus, 

in 

contempia-  if  a  person  in  contemplating  a  rose  should  make 
the  scent  a  distinct  object  of  attention,  and  lay 
aside  all  thought  of  the  form,  color,  &c.,  he 
would  draw  off,  or  abstract  that  particular  part  ; 

the  process 

or  drawing  and  therefore  employ  the  faculty  of  abstraction. 
He  would  also  employ  the  same  faculty  in  con 
sidering  whiteness,  softness,  virtue,  existence,  as 
entirely  separate  from  particular  objects. 

§  13.    The   term  abstraction,  is   also  used   to 
The  term    denote  tne  operation  of  abstracting  from  one  or 

Abstraction,  mOre  things  the  particular  part  under  consider- 
how  used. 

ation  ;  and  likewise  to  designate  the  state  of  the 

mind  when  occupied  by  abstract  ideas.     Hence, 
abstraction  is  used  in  three  senses  : 
Abstraction       lst-    To   denote    a   faculty  or   power  of  the 


2d-  To  denote  a  process  of  the  mind  ;  and, 


GENERALIZATION. 


Generalize  §  l4'  Generalization  is  the  process  of  con 
templating  the  agreement  of  several  objects  in 
certain  points  (that  is,  abstracting  the  circum 
stances  of  agreement,  disregarding  the  differ- 


CHAP.   I.] 


TERMS. 


33 


ences),  and  giving  to  all  and  each  of  these  ob-  ofseveral 
jects  a  name  applicable  to  them  in  respect  to  thing8- 
this  agreement.  For  example  ;  we  give  the 
name  of  triangle,  to  every  rectilineal  figure  hav 
ing  three  sides  :  thus  we  abstract  this  property 
from  all  the  others  (for,  the  triangle  has  three 
angles,  may  be  equilateral,  or  scalene,  or  right- 
angled),  and  name  the  entire  class  from  the  prop 
erty  so  abstracted.  Generalization  therefore 
necessarily  implies  abstraction ;  though  abstrac 
tion  does  not  imply  generalization. 


Generaliza 
tion 


implies 
abstraction. 


A  term. 


TERMS  -  SINGULAR  TERMS  -  COMMON  TERMS. 

§  15.  An  act  of  apprehension,  expressed  in 
language,  is  called  a  Term.  Proper  names,  or 
any  other  terms  which  denote  each  but  a  single 
individual,  as  "  Caesar,"  "  the  Hudson,"  "  the 
Conqueror  of  Pompey,"  are  called  Singular  singular  . 

terms. 

Terms. 

On  the  other  hand,  those  terms  which  denote 
any  individual  of  a  whole  class  (which  are  form 
ed  by  the  process  of  abstraction  and  generaliza 
tion),  are  called  Common  or  general  Terms.  For  common 
example  ;  quadrilateral  is  a  common  term,  appli 
cable  to  every  rectilineal  plane  figure  having 
four  sides  ;  River,  to  all  rivers  ;  and  Conqueror, 
to  all  conquerors.  The  individuals  for  which  a 
common  term  stands,  are  called  its  Significates. 

3 


terms. 


34  LOGIC.  [BOOK  i. 


CLASSIFICATION. 

6  16    Common  terms  afford  the  means  of  clas- 

Classiflcation. 

sification ;  that  is,  of  the  arrangement  of  objects 
into  classes,  with  reference  to  some  common  and 
distinguishing  characteristic.  A  collection,  com 
prehending  a  number  of  objects,  so  arranged,  is 
Genus,  called  a  Genus  or  Species — genus  being  the 

Bpecies. 

more  extensive  term,  and  often  embracing  many 
species. 

Exam  lea        ^or  example :   animal  is    a  genus  embracing 
in        every  thing  which  is  endowed  with  life,  the  pow- 

classification. 

er  of  voluntary  motion,  and  sensation.  It  has 
many  species,  such  as  man,  beast,  bird,  &c.  If 
we  say  of  an  animal,  that  it  is  rational,  it  !><•- 
longs  to  the  species  man,  for  this  is  the  charac 
teristic  of  that  species.  If  we  say  that  it  has 
.  wings,  it  belongs  to  the  species  bird,  for  this,  in 

like  manner,  is  the  characteristic  of  the  specLs 
bird. 

A  species  may  likewise  be  divided  into  classes, 

Subspecies 

or  or  subspecies  ;  thus  the  species  man,  may  be 
divided  into  the  classes,  male  and  female,  tind 
these  classes  may  be  again  divided  until  we  reach 
the  individuals. 

Principles        §  17.  Now,  it  will  appear  from  the  principles 
classification,  which  govern  this  system  of  classification,  that 


CHAP.   I.] 


CLASSIFICATION. 


35 


the  characteristic  of  a  genus  is  of  a  more  exten-  Genus  more 

. ~  ,  ,  ,,  extensive 

sive   signification,    but   involves   fewer   particu-  than  species, 
lars  than  that  ol  a  species.     In  like  manner,  the 
characteristic  of  a  species  is  more  extensive,  but 
less  full  and  complete,  than  that  of  a  subspecies  but  less  ful1 

and 

or  class,  and  the  characteristics  of  these  less  full    complete, 
than  that  of  an  individual. 

For  example  ;  if  we  take  as  a  genus  the  Quadri 
laterals  of  Geometry,  of  which  the  characteristic      / i\ 

is,  that  they  have  four  sides,  then  every  plane    *- * 

rectilineal  figure,  having  four  sides,  will  fail  under 
this  class.  If,  then,  we  divide  all  quadrilaterals  /  2  \ 
into  two  species,  viz.  those  whose  opposite  sides, 
taken  two  and  two,  are  not  parallel,  and  those 
whose  opposite  sides,  taken  two  and  two,  are 
parallel,  we  shall  have  in  the  first  class,  all  irreg 
ular  quadrilaterals,  including  the  trapezoid  (1  and 
2)  ;  and  in  the  other,  the  parallelogram,  the  rhom 
bus,  the  rectangle,  and  the  square  (3,4,  5,  and  6). 

If,  then,  we  divide  the  first  species  into  two 
subspecies  or  classes,  we  shall  have  in  the  one,  the 
irregular  quadrilaterals  (1),  and  in  the  other,  the 
trapezoids  (2)  ;  and  each  of  these  classes,  being- 
made  up  of  individuals  having  the  same  char 
acteristics,  are  not  susceptible  of  further  division. 
If  we  divide  the  second  species  into  two 
classes,  arranging  those  which  have  oblique  an 
gles  in  the  one,  and  those  which  have  right 


36 


LOGIC. 


[BOOK  i. 


and 


angles  in  the  other,  we  shall  have  in  the  first, 
two  varieties,  viz.  the  common  parallelogram 
and  the  equilateral  parallelogram  or  rhombus  (3 
and  4)  ;  and  in  the  second,  two  varieties  also, 
viz.  the  rectangle  and  the  square  (5  and  6). 

Now,  each  of  these    six    figures  is  a  quadn- 

vfduti famng  lateral;  and  hence,  possesses  the  characteristic 

under  the      ^  ^  genus ;  and  each  variety  of  both  species 

genus  enjoys 

aii  the      enjoys  all  the  characteristics  of  the  species  to 

characteris 
tics,        which  it  belongs,  together  with  some  other  dis 
tinguishing  feature ;  and  similarly,  of  all  classi 
fications. 

§  18.  In  special  classifications,  it  is  often  not 

necessary  to  begin  with  the  most  general  char- 

_  .  ..        acteristics;  and  then  the  genus  with  which  we 

Subaltern 

begin,  is  in  fact  but  a  species  of  a  more  extended 
classification,  and  is  called  a  Subaltern  Genus. 

For  example ;  if  we  begin  with  the  genus  Par 
allelogram,  we  shall  at  once  have  two  species, 
viz.  those  parallelograms  whose  angles  are  oblique 
and  those  whose  angles  are  right  angles ;  and  in 
each  species  there  will  be  two  varieties,  viz.  in  the 
first,  the  common  parallelogram  and  the  rhom 
bus  ;  and  in  the  second,  the  rectangle  and  square. 


genus. 


Parallelo 
gram. 


Highest 


§  19.    A   genus  which  cannot   be  considered 
as  a  species,  that   is,  which  cannot  be  referred 


CHAP.   I.]  NATURE     OF     COMMON     TERMS.  37 


to  a  more  extended  classification,  is  called  the     Highest 

.  .  genus. 

highest  genus ;  and  a  species  which  cannot  be 

Lowest 

considered  as  a  genus,  because  it  contains  only     species, 
individuals   having   the   same   characteristic,    is 
called  the  lowest  species. 

NATURE  OF  COMMON  TERMS. 

$  20.  It  should  be  steadily  kept  in  mind,  that 
the  "common  terms"  employed  in  classification,   Acommon 
have  not,  as  the  names  of  individuals  have,  any     termh<18 

*    no  real  thing 

real  existing  thing  in  nature    corresponding  to  corresP°nd- 

them;  but  that  each  is  merely  a  name  denoting 

a   certain    inadequate    notion  which   our  minds    inadequate 

have  formed  of  an  individual.     But  as  this  name 

does  not  include  any  thing  wherein  that  indi-     does  not 

include  any 

vidual  differs  from  others  of  the  same  class,  it     thing  in 
is  applicable  equally  well  to  all  or  any  of  them.    ind*vidualg 
Tims,  quadrilateral  denotes   no  real  thing,  dis-       dLffer; 
tinct  from  each  individual,  but  merely  any  recti 
lineal  figure  of  four  sides,  viewed  inadequately ; 
that  is,   after  abstracting  and  omitting  all   that 
is  peculiar  to  each  individual  of  the  class.     By 

J          but  is 

this  means,  a  common  term  becomes  applicable  applicable  to 

many 

alike  to  any  one  of  several  individuals,  or,  taken  individuals, 
in  the  plural,  to  several  individuals  together. 

Much  needless  difficulty  has  been  raised  re-  Needless 
specting  the  results  of  this  process  :  many  hav-  difficulty- 
ing  contended,  and  perhaps  more  having  taken 


38  LOGIC.  [BOOK  i. 


the  interpre 
tation  of 
common 
terms. 


Difficulty  in  it  for  granted,  that  there  must  be  some  really 
existing  thing  corresponding  to  each  of  those 
common  terms,  and  of  which  such  term  is  the 
name,  standing  for  and  representing  it.  For  ex 
ample  ;  since  there  is  a  really  existing  thing  cor- 
Noone  responding  to  and  signified  by  the  proper  and 

real  thing    smgu|ar  name   «  yEtna,"   it   has   been    supposed 

correspond-  o 

in<?  to  each.  faat  fae  common  term  "  Mountain"  must  have 
some  one  really  existing  thing  corresponding  to 
it,  and  of  course  distinct  from  each  individual 
mountain,  yet  existing  in  each,  since  the  term, 
being  common,  is  applicable,  separately,  to  every 
one  of  them. 

The  fact  is,  the  notion  expressed  by  a  common 

term  is  merely   an   inadequate    (or  incomplete) 

inadequate    n°ti°n  of  an  individual  ;  and  from  the  very  cir- 

on  par-   cumsf;ance  of  its  inadequacy,  it  will  apply  equally 

signaling     wej}  ^o  any  one  of  several  individuals.     For  ex- 

the  thing. 

ample  ;  if  I  omit  the  mention  and  the  consider 
ation  of  every  circumstance  which  distinguishes 
yEtna  from  any  other  mountain,  I  then  form  ^^ 
notion,  that  inadequately  designates  /Etna.  This 
u  Mountain"  notion  is  expressed  by  the  common  term  "  moun- 
tam'"  which  does  not  imply  any  of  the  peculiar- 


toa11       ities  of  the  mountain  /Etna,  and  is  equally  ap- 

moun  tains. 

plicable  to  any  one  of  several  individuals. 

In  regard  to  classification,  we  should  also  bear 
in  mind,  that  we    may   fix,    arbitrarily,    on    the 


CHAP.   I.]  SCIENCE.  39 

characteristic  which  we  choose  to  abstract  and    May  fix  on 

attributes 

consider  as  the  basis  of  our  classification,  disre-    arbitrarily 
garding  all  the  rest :  so  that  the  same  individual  classik°cr.ltion 
may  be  referred  to  any  of  several  different  spe 
cies,  and  the  same  species  to  several  genera,  as 
suits  our  purpose. 

SCIENCE. 

§  21.    Science,   in    its    popular    signification, 
means  knowledge.*     In  a  more  restricted  sense,     science 

in  its  general 

it  means  knowledge  reduced  to  order;  that  is,      sense. 
knowledge  so  classified  and  arranged  as  to  be 
easily  remembered,  readily  referred  to,  and  ad-      Hasa 
vantageously   applied.       In    a    more    strict    and  g^oition. 
technical  sense,  it  has  another  signification. 

"  Every  thing  in  nature,   as    well  in  the  in-     yiewg  of 
animate   as  in  the  animated  world,   happens  or       Kaut- 
is  done  according  to  rules,   though  we  do  not 
always   know  them.     Water  falls   according  to 
the  laws  of  gravitation,  and  the  motion  of  walk-  Generallaws- 
ing  is  performed  by  animals  according  to  rules. 
The  fish  in  the  water,  the  bird  in  the  air,  move 
according  to  rules.     There  is  nowhere  any  want 
of  rule.     When  we  think  we  find  that  want,  we 

Nowhere 

can  only  say  that,  in  this  case,  the  rules  are  un-   any  want  of 

rule. 

known  to  us.  f 

Assuming  that  all   the  phenomena  of  nature 

*  Section  23.  f  Kant. 


40  LOGIC.  [BOOK  i. 

science     are  consequences  of  general  and  immutable  laws, 

a  technical    we  may  define   Science    to   be  the    analysis    of 

sense  defined:  ^^  jawg> — comprehending  not  only  the   con- 

an  analysis    nected  processes   of  experiment    and  reasoning 

of  the  laws 

of  nature,  which  make  them  known  to  man,  but  also  those 
processes  of  reasoning  which  make  known  their 
individual  and  concurrent  operation  in  the  de 
velopment  of  individual  phenomena. 

ART. 

§  22.  Art  is  the  application  of  knowledge  to 
Arti       practice.     Science  is  conversant  about  knowl- 
appucation   e(}ge .  Art  is  the  use  or  application  of  knowl- 
science,     edge,  and  is  conversant  about  works.     Science 
has  knowledge  for  its  object :  Art  has  knowledge 
for  its  guide.     A  principle  of  science,  when  ap 
plied,  becomes  a  rule  of  art.     The  developments 
of  science  increase  knowledge :  the  applications 
and       of  art  add  to  works.     Art,  necessarily,  presup- 
prosupposea  p0ses  knowledge :  art,  in  any  but  its  infant  state, 

knowledge.    r  J 

presupposes  scientific  knowledge ;  and  if  every 
art  does  not  bear  the  name  of  the  science  on 
which  it  rests,  it  is  only  because  several  sciences 
are  often  necessary  to  form  the  groundwork  of 
a  single  art.  Such  is  the  complication  of  hu- 

Many  things  „  . 

must  be     man  anairs,  that  to  enable  one  thing  to  be  done, 

ibreTn^ctn  il:  is  often  recluisite  to  know  the  nature  and  prop- 
be  done,     erties  of  many  things. 


CHAP.   II.]  KNOWLEDGE.  41 


CHAPTER    II. 


SOURCES  AND  MEANS  OF  KNOWLEDGE — INDUCTION. 


KNO  WLEGDE. 

§  23.  KNOWLEDGE  is  a  clear  and  certain  con-  Knowledge 

a  clear  con 
ception  of  that  which  is  true,  and  implies  three    ceptionof 

things:  what  is  true: 

1st.  Firm  belief;    2d.  Of  what  is   true;    and,    implies— 

~,     .  ,  1st.   Firm 

3d.  On  sufficient  grounds.  belief . 

If  any  one,  for  example,  is  in  doubt  respecting  *&.  of  what 

is  true ; 

one  of  Legendre's   Demonstrations,   he   cannot     3d.  on 
be  said  to  know  the  proposition  proved  by  -it.     If,      roim^ 
again,  he  is  fully  convinced  of  any  thing  that  is 
not  true,  he  is  mistaken  in  supposing  himself  to 

Examples. 

know  it;  and  lastly,  if  two  persons  are  each/^% 
confident,  one  that  the  moon  is  inhabited,  and 
the  other  that  it  is  not  (though  one  of  these 
opinions  must  be  true),  neither  of  them  could 
properly  be  said  to  know  the  truth,  since  he 
cannot  have  sufficient  proof  of  it. 


42  LOGIC.  [BOOK  I. 

FACTS     AND     TRUTHS. 

Knowledge  is       §  24'  ®UT  knowledge  is  of  two  kinds  :  of  facts 
of  facts  and   and  truths.     A  fact  is  any  thing  that  HAS  BEEN 

truths.  J 

or  is.  That  the  sun  rose  yesterday,  is  a  fact : 
that  he  gives  light  to-day,  is  a  fact.  That  wa 
ter  is  fluid  and  stone  solid,  are  facts.  We  de 
rive  our  knowledge  of  facts  through  the  medium 
of  the  senses. 
Truth  an  Truth  is  an  exact  accordance  with  what  HAS 

accordance 

with  what    BEEN,  is,  or  SHALL  BE.     There  are  two  methods 

has  been,  is,       ,-.  .     . 

or  shaii  be.   oi  ascertaining  truth  : 

ofanrtri^      lst'    %  comparing   known   facts   with   each 
ing  it.       other;  and, 

2dly.  By  comparing  known  truths  with  each 
other. 

Hence,  truths  are  inferences  either  from  facts 
or  other  truths,  made  by  a  mental  process  called 
Reasoning. 

§  25.  Seeing,  then,  that  facts  and  truths  are  the 
elements  of  all  our  knowledge,  and  that  knowl 
edge  itself  is  but  their  clear  apprehension,  their 
knowledge,  firm  belief,  and  a  distinct  conception  of  their 
relations  to  each  other,  our  main  inquiry  is,  How 
are  we  to  attain  unto  these  facts  and  truths, 
which  are  the  foundations  of  knowledge  ? 

1st.  Our  knowledge  of  facts  is  derived  through 


CHAP.   II.]  FACTS     AND     TRUTHS.  43 

the  medium  of  our  senses,  by  observation,  exper 
iment,*  and  experience.     We  see  the  tree,  and     HOW  we 

arrive  at  a 

perceive  that  it  is  shaken  by  the  wind,  and  note  knowledge  of 
the  fact  that  it  is  in  motion.  We  decompose 
water  and  find  its  elements ;  and  hence,  learn 
from  experiment  the  fact,  that  it  is  not  a  simple 
substance.  We  experience  the  vicissitudes  of 
heat  and  cold;  and  thus  learn  from  experience 
that  the  temperature  is  not  uniform. 

The  ascertainment  of  facts,  in  any  of  the  ways 
above  indicated,  does  not  point  out  any  connec-  This  does  not 

point  out  a 

tion  between  them.     It  merely  exhibits  them  to   connection 
the  mind  as  separate  or  isolated;  that  is,  each     bt^en 
as   standing   for   a   determinate   thing,   whether 
simple    or  compound.     The   term   facts,  in    the 
sense  in  which  we  shall  use  it,  will   designate 
facts  of  this  class  only.     If  the  facts  so  ascer 
tained  have  such  connections  with  each  other,  when  they 
that  additional  facts  can  be  inferred  from  them,  nectiou  that 

,  .     c  •  .     ,      -,  ,     i         ,j  .          is  pointed  out 

that  inference  is  pointed  out  by  the  reasoning   bytherea. 
process,  which  is  carried  on,  in  all  cases,  by  com 
parison. 

2dly.  A  result  obtained  by  comparing  facts,  we  Truth,  found 
have  designated  by  the   term   Truth.      Truths, 
therefore,  are  inferences  from  facts  ;  and  every 

*  Under  this  term  we  include  all  the  methods  of  inves 
tigation  and  processes  of  arriving  at  facts,  except  the  pro 
cess  of  reasoning. 


44  LOGIC.  [BOOK  i. 


and       truth  has  reference  to  all  the  singular  facts  from 

inferred 
from  them. 


which  it  is  inferred.     Truths,  therefore,  are  re 


sults  deduced  from  facts,  or  from  classes  of  facts. 
Such  results,  when  obtained,  appertain  to  all  facts 
of  the  same  class.  Facts  make  a  genus  :  truths, 
a  species ;  with  the  characteristic,  that  they  be 
come  known  to  us  by  inference  or  reasoning. 

HOW  §  26.    How,   then,   are   truths  to  be  inferred 

truths  •**--,. 

inferred  from  irom  tacts   by  the  reasoning  process?      There 

facts  by  the 
reasoning 

process.         js^  When  the  instances  are  so  few  and  simple 

ist  case,     that  the  mind  can  contemplate  all  the  facts  on 

which  the  induction  rests,  and  to  which  it  refers, 

and  can  make  the  induction  without  the  aid  of 

other  facts  ;  and, 

2dly.  When  the  facts,  being  numerous,  com 
plicated,  and  remote,  are  brought  to  mind  only 
by  processes  of  investigation. 

INTUITIVE     TRUTH. 

§  27.   Truths  which  become  known  by  con- 
sidering  all  the  facts  on  which  they  depend,  and 
which   are   inferred   the  moment  the  facts   are 
truths,      apprehended,  are  the   subjects  of  Intuition,  and 
are  called  Intuitive  or  Self-evident  Truths.     The 
intuition     term  Intuition  is  strictly  applicable  only  to  that 
mode   of  contemplation   in  which    we   look   at 


CHAP.   II.]  INTUITIVE     TRUTH.  45 


facts,  or  classes  of  facts,  and  apprehend  the 
relations  of  those  facts  at  the  same  time,  and 
by  the  same  act  by  which  we  apprehend  the 
facts  themselves.  Hence,  intuitive  or  self-evi-  HOW  intuitive 

...          truths  are 

dent  truths   are  those  which   are  conceived  in  conceived  in 
the  mind  immediately  ;    that  is,  which  are  per 
fectly  conceived  by   a  single  process  of  induc 
tion,  the  moment  the  facts  on  which  they  depend 
are    apprehended,   without   the    intervention   of 
other  ideas.    They  are  necessary  consequences  of 
conceptions  respecting  which  they  are  asserted.    Axioms  of 
The  axioms  of  Geometry  afford  the  simplest  and  Je^pies? 
most  unmistakable  class  of  such  truths.  kimU 

"A  whole  is  equal  to  the  sum  of  all  its  parts,"     A  whole 

r  ,      .     c  c  equal  to  the 

is  an  intuitive  or  sen-evident  truth,  interred  from    Slimofaii 


facts  previously  learned.     For  example  ;  having 

learned  from  experience  and  through  the  senses       tmth* 

what  a  whole  is,  and,  from  experiment,  the  fact 

that  it  may  be  divided  into  parts,  the  mind  per 

ceives  the  relation  between  the  whole  and  the 

sum  of  the  parts,  viz.  that  they  are  equal  ;  and 

then,  by  the   reasoning  process,  infers  that  the  HOW  interred. 

same    will   be    true    of  every  other   thing;  and 

hence,   pronounces    the   general   truth,  that   "a 

whole  is  equal  to  the  sum  of  all  its  parts."     Here 

all  the  facts  from  which  the  induction  is  drawn,  A11  the  facts 

are  presented 

are  presented  to   the   mind,   and  the   induction  to  the  mind. 
is  made  without  the  aid  of  other  facts  ;  hence, 


46  LOGIC.  [BOOK  i. 

AH  the     it  is  an  intuitive  or  self-evident  truth.     All  the 

axioms  are  . 

deduced  i»    other   axioms   of  Geometry    are    deduced   from 

the  same  .  ,    ,  r   •     r- 

way.  premises  and  by  processes  of  inference,  entirely 
similar.  We  would  not  call  these  experimental 
truths,  for  they  are  not  alone  the  results  of  ex 
periment  or  experience.  Experience  and  exper 
iment  furnish  the  requisite  information,  but  the 
reasoning  power  evolves  the  general  truth. 

"  When  we  say,  the  equals  of  equals  are  equal, 
we  mentally  make  comparisons  in  equal  spaces, 

These      equal  times,   &c. ;    so  that  these   axioms,   how- 
axioms  are 
general     ever  self-evident,  are   still  general  propositions : 

so  far  of  the  inductive  kind,  that,  independently 

of  experience,  they  would  not  present  themselves 

to  the  mind.     The  only  difference  between  these 

and  axioms  obtained  from  extensive  induction  is 

Difference    this  i   that,  in  raising  the   axioms  of  Geometry, 

themtnd    the  instances  offer  themselves  spontaneously,  and 

rrop^itions,  without  the  trouble  of  search,  and  are  few  and 

qrtrediifert  simP^e  :  in  raising  those  of  nature,  they  are  in- 

research.     finitely  numerous,  complicated,  and  remote;   so 

that  the  most  diligent  research  and  the  utmost 

acuteness  are  required  to  unravel  their  web,  and 

place  their  meaning  in  evidence."* 

*  Sir  John  Herschel's  Discourse  on  the  study  of  Natural 
Philosophy. 


CHAP.   II.]  LOGICAL     TRUTHS.  47 

TRUTHS,     OR     LOGICAL     TRUTHS. 

§  28.  Truths  inferred  from  facts,  by  the  process 
of  generalization,  when  the  instances  do  not  offer      Truths 

.          generalized 

themselves  spontaneously  to  the  mind,  but  require  from  factS5 
search  and  acuteness  to  discover  and  point  out  tra^8din_ 
their  connections,  and  all  truths  inferred  from  ferredfrom 

truths. 

truths,  might  be  called  Logical  Truths.  But  as 
we  have  given  the  name  of  intuitive  or  self- 
evident  truths  to  all  inferences  in  which  all  the 
facts  were  contemplated,  we  shall  designate  all 
others  by  the  simple  term,  TRUTHS. 

It  might  appear  of  little  consequence  to  dis-  Necessityof 
tinffuish   the   processes   of  reasoning    by  which   thedistinc- 

J  tion,  being 

truths  are  inferred  from  facts,  from  those  in  which  the  basis  of  a 

classification. 

we  deduce  truths  from  other  truths ;  but  this  dif 
ference  in  the  premises,  though  seemingly  slight, 
is  nevertheless  very  important,  and  divides  the 
subject  of  logic,  as  we  shall  presently  see,  into 
two  distinct  and  very  different  branches. 

LOGIC. 

§  29.   Logic  takes  note  of  and  decides  upon      Logic 
the  sufficiency  of  the  evidence  by  which  truths  su^cieesncyeof 
are  established.     Our  assent  to  the  conclusion    evidence. 
being  grounded  on  the  truth  of  the  premises,  we 
never  could   arrive   at   any  knowledge  by  rea 
soning,   unless    something  were  known    antece 
dently  to   all  reasoning.     It  is  the  province  of  its  province. 


48  LOGIC.  [BOOK  i. 


Furnishes    Logic  to  furnish  the  tests  by  which  all  truths 
troth.       that  are  not  intuitive  may  be  inferred  from  the 
premises.     It  has  nothing  to  do  with  ascertain 
ing  facts,  nor  with  any  proposition  which  claims 
to   be   believed   on  its  own  intrinsic  evidence ; 
that  is,  without  evidence,  in  the  proper  sense  of 
Has  nothing  the  word.     It  has  nothing  to  do  with  the  original 
intuitive  pro  data,   or  ultimate  premises   of  our   knowledge ; 
positions,  nor  wit}1  ^gjj.  number  or  nature,  the  mode  in  which 

with  original 

data;  they  are  obtained,  or  the  tests  by  which  they 
are  distinguished.  But,  so  far  as  our  knowledge 
is  founded  on  truths  made  such  by  evidence, 

but  supplies 

aii  tests  for   that  is,  derived  from  facts  or  other  truths  pre- 

general 

propositions,  viously  known,  whether  those  truths  be  particu 
lar  truths,  or  general  propositions,  it  is  the  prov 
ince  of  Logic  to  supply  the  tests  for  ascertaining 
the  validity  of  such  evidence,  and  whether  or 
not  a  belief  founded  on  it  would  be  well  ground- 

o 

ed.     And  since  by  far  the  greatest  portion  of 
The  greatest  our  knowledge,  whether  of  particular  or  general 

portion  of  our 

knowledge  truths,  is  avowedly  matter  of  inference,  nearly 
the  wn°le>  not  only  of  science,  but  of  human 
conduct,  is  amenable  to  the  authority  of  logic. 


towhat 


CHAP.  II.]  INDUCTION.  49 


IN  DUCTION. 

§  30.  That  part  of  logic  which  infers  truths 
from  facts,  is  called  Induction.     Inductive  rea 
soning  is   the  application  of  the  reasoning  pro- 
reasoning 

cess  to  a  given  number  of  facts,  for  the  purpose   applicable, 
of  determining  if  what  has  been  ascertained  re 
specting  one  or  more  of  the  individuals  is  true 
of  the  whole  class.      Hence,   Induction  is  not    induction 

defined. 

the  mere  sum  of  the  facts,  but  a  conclusion 
drawn  from  them. 

The   logic  of   Induction  consists  in   classing    Logic  of 

Induction. 

the  facts  and  stating  the  inference  in  such  a 
manner,  that  the  evidence  of  the  inference  shall 
be  most  manifest. 


§  31.  Induction,  as  above  defined,  is  a  process  induction 
of  inference.  It  proceeds  from  the  known  to  f^the 
the  unknown;  and  any  operation  involving  no  knowntothe 

*  unknown. 

inference,  any  process  in  which  the  conclusion 
is  a  mere  fact,  and  not  a  truth,  does  not  fall 

within  the  meaning  of  the  term.     The  conclu-  The  conclu 
sion  broader 

sion  must  be  broader  than  the  premises.     The     than  the 

premises. 

premises  are  facts  :  the   conclusion   must  be   a 
truth. 

Induction,  therefore,  is  a  process  of  general-    induction, 

.  a  process  of 

ization.     It   is   that   operation   of  the   mind  by   generaiiza- 
which  we  infer  that  what  we  know  to  be  true 

4 


59 


LOGIC.  [BOOK  i. 


in  which  in  a  particular  case  or  cases,  will  be  true  in  all 
^h^u!'  cases  whicn  resemble  the  former  in  certain  as- 
true  under  signable  respects.  In  other  words,  Induction  is 

particular 

circumstan-   the  process  by  which  we  conclude    that  what 

ccs  will  be  .     .          ,         r  , 

euniver-  is  true  of  certain  individuals  of  a  class  is  true 
8aUy<       of  the  whole  class ;  or  that  what  is  true  at  cer 
tain  times,  will  be  true,  under  similar  circum 
stances,  at  all  times. 


Induction 


j  32.  Induction  always  presupposes,  not  only 
presupposes  fa^  faQ  necessary  observations  are  made  with 

accurate  and 

necessary    the  necessary  accuracy,  but  also  that  the  results 

observations.  _  -11 

of  these  observations  are,  so  far  as  practicable, 
connected  together  by  general  descriptions  :  ena 
bling  the  mind  to  represent  to  itself  as  wholes, 
whatever  phenomena  are  capable  of  being  so 
represented. 

To  suppose,  however,   that  nothing  more  is 

More  is     required  from  the  conception  than  that  it  should 

necessary     serve  to  connect  the  observations,  would  be  to 

than  to 

connect  the  substitute    hypothesis   for  theory,   and    imasnna- 

observations: 

we  must     tion  for  proof.     The  connecting  link  must   be 

infer  from  .  . 

them.  some  character  which  really  exists  in  the  iacts 
themselves,  and  which  would  manifest  itself 
therein,  if  the  condition  could  be  realized  which 
our  organs  of  sense  require. 

For  example ;  Blakewell,  a  celebrated  English 
cattle-breeder,  observed,  in  a  great   number  of 


CHAP.  II.]  INDUCTION.  51 


individual  beasts,  a  tendency  to  fatten  readily,    Example  of 
and  in  a  great  number  of  others  the  absence  of  the  English 


this  constitution  :  in  every  individual  of  the  for- 
mer  description,  he  observed  a  certain  peculiar 
make,  though  they  differed  widely  in  size,  color, 
&c.  Those  of  the  latter  description  differed  no 
less  in  various  points,  but  agreed  in  being  of  a 
different  make  from  the  others.  These  facts  were  HOW  he 
his  data  ;  from  which,  combining  them  with  the 


general  principle,  that  nature  is  steady  and  uni-     whyhe 
form  in  her  proceedings,  he  logically  drew  the 
conclusion  that  beasts  of  the  specified  make  have 
universally  a  peculiar  tendency  to  fattening. 

The  principal  difficulty  in  this  case  consisted  in  what  the 

,  .  .  difficulty 

in  making  the  observations,  and  so  collating  and  consisted. 
combining  them  as  to  abstract  from  each  of  a 
multitude  of  cases,  differing  widely  in  many  re 
spects,  the  circumstances  in  which  they  all 
agreed.  But  neither  the  making  of  the  observa 
tions,  nor  their  combination,  nor  the  abstraction, 
nor  the  judgment  employed  in  these  processes, 
constituted  the  induction,  though  they  were  all 
preparatory  to  it.  The  Induction  consisted  in  in  what  the 
the  generalization  ;  that  is,  in  inferring  from  all 
the  data,  that  certain  circumstances  would  be 
found  in  the  whole  class. 

The  mind  of  Newton  was  led  to  the  universal 
law,  that  all  bodies  attract  each  other  by  forces 


52  LOGIC.  [BOOK  i. 

Newton's  varying  directly  as  their  masses,  and  inversely 
*&****<*  as  tne  sauares  °f  tneir  distances,  by  Induction, 
universal  jje  saw  an  appie  falling  frOm  the  tree :  a  mere 

gravitation. 

fact ;  and  asked  himself  the  cause ;  that  is,  if  any 
inference  could  be  drawn  from  that  fact,  which 
should  point  out  an  invariable  antecedent  condi- 
HOW  he  tion.  This  led  him  to  note  other  facts,  to  prose 
cute  experiments,  to  observe  the  heavenly  bodies, 


their       until   from   many    facts,    and    their   connections 

connections. 

with  each  other,  he  arrived  at  the  conclusion, 
that  the  motions  of  the  heavenly  bodies  were  gov 
erned  by  general  laws,  applicable  to  all  matter ; 
that  the  stone  whirled  in  the  sling  and  the  earth 
rolling  forward  through  space,  are  governed  in 
their  motions  by  one  and  the  same  law.  He 
The  use  then  brought  the  exact  sciences  to  his  aid,  and 

which  he 

made  of     demonstrated  that  this  law  accounted  for  all  the 
science,     phenomena,  and  harmonized  the  results  of  all  ob 
servations.      Thus,   it  was   ascertained  that  the 
what  was    laws  which  regulate   the  motions  of  the  heav- 
5Ult'    enly   bodies,    as   they   circle   the   heavens,    also 
guide  the  feather,  as  it  is  wafted  along  on  the 
passing  breeze. 


The  ways  of       §  33.  We  have  already  indicated  the  ways  in 

ascertaining  . 

facts  are  which  the  lacts  are  ascertained  from  which  the 
inferences  are  drawn.  But  when  an  inference 
can  be  drawn  ;  how  many  facts  must  enter  into 


CHAP.   II.]  INDUCTION.  53 


the  premises  ;  what  their  exact  nature  must  be  ;      but  we 
and  what  their  relations  to  each  other,  and  to 


the  inferences  which  flow  from  them  ;  are  ques-   in  a11  cases' 

when  we  can 

tions  which  do  not  admit  of  definite  answers,     draw  on 

inference. 

Although  no  general  law  has  yet  been  discov 
ered  connecting  all  facts  with  truths,  yet  all  the       No 

general  law. 

uniformities  which  exist  in  the  succession  of  phe 
nomena,  and  most  of  those  which  prevail  in  their 
coexistence,  are  either  themselves  laws  of  cau 
sation  or  consequences  resulting  and  corollaries 
capable  of  being  deduced  from,  such  laws.  It 
being  the  main  business  of  Induction  to  deter-  Businesg 
mine  the  effects  of  every  cause,  and  the  causes  of 

Induction. 

of  all  effects,  if  we  had  for  all  such  processes 
general  and  certain  laws,  we  could  determine,     what  is 
in  all  cases,  what  causes  are  correctly  assigned 
to  what  effects,  and  what  effects  to  what  causes, 
and  we  should  thus  be  virtually  acquainted  with 
the  whole  course  of  nature.     So  far,  then,  as  we    How  far  a 
can   trace,   with   certainty,  the   connection   be-     8cience- 
tween  cause  and  effect,  or  between  effects  and 
their  causes,  to  that  extent  Induction  is  a  sci 
ence.     When  this  cannot  be  done,  the  conclu 
sions  must  be,  to  some  extent,  conjectural. 


LOGIC.  [BOOK  i. 


CHAPTER    III. 

DEDUCTION NATURE    OF   THE    SYLLOGISM ITS    USES    AND    APPLICATIONS. 

DEDUCTION. 

§  34.    WE    have   seen   that   all   processes   of 
inductive     Reasoning,  in  which  the  premises  are  particular 
Pre^nTng°    ^cts,    and   the   conclusions  general    truths,   are 
called  Inductions.     All  processes  of  Reasoning, 
in  which  'the  premises  are  general  truths  and  the 
Deductive    conclusions  particular  truths,  are  called  Deduc 
tions.     Hence,    a   deduction   is    the    process    of 
Deduction    reasoning  by  which  a  particular  truth  is  inferred 
from  other  truths  which  are  known  or  admitted. 
Deductive     The  formula  for  all  deductions  is  found  in  the 
Syllogism,  the  parts,  nature,  and 
we  shall  now  proceed  to  explain. 


Syllogism,  the  parts,  nature,  and  uses  of  which 


PROPOSITIONS. 


Proposition,  §  35-  A  proposition  is  a  judgment  expressed 
in  words.  Hence,  a  proposition  is  defined  logi 
cally,  "  A  sentence  indicative  :"  affirming  or 


*  Section  30. 


CHAP.  III.]  PROPOSITIONS.  55 

denying;  therefore,  it  must  not  be  ambiguous,  must  not  be 

ambiguous ; 

for  that  which  has   more  than  one  meaning  is   norimpor. 
in    reality  several   propositions ;    nor  imperfect,  j^™^ 
nor  ungrammatical,  for  such   expressions  have 
no  meaning  at  all. 

§  36.    Whatever  can  be  an  object  of  belief, 
or  even  of  disbelief,  must,  when  put  into  words,  A  proposition 
assume  the  form  of  a  proposition.     All  truth  and 
all  error  lie  in  propositions.     What  we   call  a 
truth,  is  simply  a  true  proposition;   and  errors  its  nature,— 

extent. 

are  false  propositions.  To  know  the  import  of 
all  propositions,  would  be  to  know  all  questions 
which  can  be  raised,  and  all  matters  which  are  Embracesa11 

truth  and  all 

susceptible   of  being   either   believed   or   disbe-       error, 
lieved.     Since,  then,  the  objects  of  all  belief  and 
all  inquiry  express  themselves  in  propositions,  a 

sufficient  scrutiny  of  propositions  and  their  va-  An  examina 
tion  of 
rieties  will  apprize  us  of  what  questions  mankind  propositions 

have  actually  asked  themselves,  and  what,  in  the 
nature  of  answers  to  those  questions,  they  have 
actually  thought  they  had  grounds  to  believe. 


§  37.  The  first  glance  at  a  proposition  shows  A  proposition 

,          .      .       f.  ,    ..  .  ,  is  formed  by 

that  it  is  formed  by  putting  together  two  names.  putting  two 
Thus,  in  the  proposition,  "Gold  is  yellow,"  the 


names 
together. 

property  yellow  is  affirmed  of  the  substance  gold. 
In  the  proposition,  "  Franklin  was   not  born  in 


56 


LOGIC. 


[BOOK  i 


England,"  the  fact  expressed  by  the  words  born 
in  England  is  denied  of  the  man  Franklin. 

§  38.    Every   proposition    consists    of    three 
parts  :  the  Subject,  the  Predicate,  and  the  Co- 

subet,     Pu^a-      ^ne  su^Ject   *s  tne  name   denoting   the 
Predicate,    person  or  thing  of  which  something  is  affirmed 

and 

copula,  or  denied  :  the  predicate  is  that  which  is  affirm 
ed  or  denied  of  the  subject  ;  and  these  two  are 
called  the  terms  (or  extremes),  because,  logically, 
the  subject  is  placed  first,  and  the  predicate  last. 
The  copula,  in  the  middle,  indicates  the  act  oi 
judgment,  and  is  the  sign  denoting  that  there  is 
ar  affirmation  or  denial.  Thus,  in  the  proposi- 

subject     ti«  n,  "  The  earth  is  round  ;"  the  subject  is   the 

defined. 

words  "  the  earth,"  being  that  of  which  some 
thing  is  affirmed  :  the  predicate,  is  the  word  round, 
which  denotes  the  quality  affirmed,  or  (as  the 

Predicate,  pb*  se  is)  predicated  :  the  word  is,  which  serves 
as  connecting  mark  between  the  subject  and 
th(  predicate,  to  show  that  one  of  them  is  af 
firmed  of  the  other,  is  called  the  Copula.  The 

mustTe  cc  ^a  must  ke  either  is,  or  is  NOT,  the  substan- 
is  or  is  NOT.  th  verb  being  the  only  verb  recognised  by 

AH  verbs     Logic.     All  other  verbs  are  -resolvable,  by  means 

resolvable         r  ,1  i  ,       ,,  , 

into  "to  be."  ol  tne  verb     to  be>    and  a  participle  or  adjective. 
For  example  : 

"  The  Romans  conquered  :" 


CHAP.  III.]  SYLLOGISM.  57 


the  word  "  conquered"  is  both  copula  and  predi-    Examples 
cate,  being  equivalent  to  "were  victorious."  Co  ula 

Hence,  we  might  write, 


"  The  Romans  were  victorious," 

in  which  were  is  the  copula,  and  victorious  the 
predicate. 


§  39.    A  proposition   being  a  portion   of  dis-  Aproposition 

is  either 

course,  in  which  something  is  affirmed  or  denied   affirmative 

c  ,  .  17  .   .  i          T     •  T     i     or  negative 

01  something,  all  propositions  may  be  divided 
into  affirmative  and  negative.  An  affirmative 
proposition  is  that  in  which  the  predicate  is  af 
firmed  of  the  subject ;  as,  "  Caesar  is  dead."  \ 
negative  proposition  is  that  in  which  the  predicate 
is  denied  of  the  subject ;  as,  "  Caesar  is  not  dead." 
The  copula,  in  this  last  species  of  proposition,  in  the  last, 

,,      ,  ,  ,,         ,  .    ,       .         ,        the  copula  is, 

consists   of  the   woras   "is   NOT,     which  is   'he      ISNOT 
sign  of  negation  ;  "  is"  being  the  sign  of  affi    .1- 
ation. 

SYLLOGISM. 

§  40.  A  syllogism  is  a  form  of  stating  the  c   .1-   A  syllogism 

consists  of 

nection  which    may    exist,   for    the    purpose    of  three  propo- 
reasoning,  between  three  propositions.      Hence, 
to    a   legitimate    syllogism,    it    is    essential   that 

J  Two  sire 

there  should  be  three,   and  only  three,  proposi-    admitted; 


58  LOGIC.  [BOOK  i. 

and  the  third  tions.     Of  these,   two  are   admitted  to  be    true, 
fromuiem.    anc^  are  called  the  premises  :  the  third  is  proved 
from  these  two,    and   is   called   the   conclusion. 
For  example  : 

"  All  tyrants  are  detestable  : 
Caesar  was  a  tyrant  ; 
Therefore,  Caesar  was  detestable." 

Now,  if  the  first  two  propositions  be  admitted, 

the  third,  or  conclusion,  necessarily  follows  from 

them,  and  it  is  proved  that  CAESAR  was  detestable. 

Major  Term       Of  the  two  terms  of  the  conclusion,  the  Predi- 

defined.     cate  (detestabie)   is   calied  the  major  term,  and 

the  Subject  (Caesar)  the  minor  term  ;  and  these 
two  terms,  together  with  the  term  "tyrant," 
make  up  the  three  propositions  of  the  syllogism, 
Minor  Term.  —  eacn  term  being  used  twice.  Hence,  every 
syllogism  has  three,  and  only  three,  different 
terms. 

premL  ^^e  premiss,  into  which  the  Predicate  of  the 
defined,  conclusion  enters,  is  called  the  major  premiss  ; 
Minor  the  other  is  called  the  minor  premiss,  and  con- 

Premiss. 


other  term,  common  to  the  two  premises,  and 
with  which  both  the  terms  of  the  conclusion  were 
separately  compared,  before  they  were  compared 


MiddleTerm. 

the  syllogism  above,   "detestable"  (in  the  con- 


CHAP.   III.] 


SYLLOGISM. 


59 


elusion)  is  the  major  term,  and  "  Caesar"  the  mi-    Example, 

pointing  out 

nor  term :  hence,  Miljor 

premiss, 


Minor 

premiss,  and 
Middle  Term. 


"  All  tyrants  are  detestable," 
is  the  major  premiss,  and 

"  Caesar  was  a  tyrant," 
the  minor  premiss,  and  "  tyrant"  the  middle  term. 

§  41.  The  syllogism,  therefore,  is  a  mere  for 
mula  for  ascertaining  what  may,  or  what  may      a  mere 
not,  be  predicated  of  a  subject.     It  accomplishes 
this  end  by  means   of  two  propositions,  viz.  by 
comparing  the   given  predicate  of  the  first   (a  HOW  applied. 
Major  Premiss),   and  the  given   subject  of  the 
second  (a  Minor  Premiss),  respectively  with  one 
and  the  same  third  term  (called  the  middle  term), 
and  thus — under  certain  conditions,  or  laws  of 
the  syllogism — to  be  hereafter  stated — eliciting 
the  truth  (conclusion)  that  the  given  predicate 
must  be  predicated  of  that  subject.     It  will  be    use  of  the 
seen  that  the   Major  Premiss  always   declares,     premiss. 
in  a  general  way,  such  a  relation  between  the 
Major  Term  and  the  Middle  Term  ;  and  the  Mi-  or  the  Minor, 
nor  Premiss  declares,  in  a  more  particular  way, 
such  a  relation  between  the  Minor  Term  and 
the  Middle  Term,  as    that,  in   the   Conclusion,      or  the 

Middle  Term. 

the  Minor  Term  must  be  put  under  the  Major 
Term  ;  or  in  other  words,  that  the  Major  Term 
must  be  predicated  of  the  Minor  Term. 


60  LOGIC.  [BOOK  i. 


ANALYTICAL  OUTLINE  OF  DEDUCTION. 

Reasoning        §  42.  In  every  instance  in  which  we  reason, 

aed'     in  the  strict  sense  of  the  word,  that  is,  make  use 

of  arguments,  whether  for  the   sake  of  refuting 

an  adversary,  or  of  conveying  instruction,  or  of 

satisfying  our  own  minds  on  any  point,  whatever 

may  be  the  subject  we  are  engaged  on,  a  certain 

process  takes  place  in  the  mind,  which  is  one 

The  process,  and  the  same  in  all  cases  (provided  it  be  cor- 

thesame.'   rectly  conducted),  whether  we  use  the  inductive 

process  or  the  deductive  formulas. 

Of  course  it  cannot  be  supposed   that  every 
Everyone    one  is  even  conscious  of  this  process  in  his  own 

not  conscious  ,  ,      , 

ofthe      mind;    much  less,  is  competent  to  explain  the 

process,     principles  on  which  it  proceeds.     This  indeed  is, 

The  same  for  and  cannot  but  be,   the  case  with  every  other 

every  other 

process.  Process  respecting  which  any  system  has  been 
formed  ;  the  practice  not  only  may  exist  inde 
pendently  of  the  theory,  but  must  have  preceded 
the  theory.  There  must  have  been  Language 
Elements  and  before  a  system  of  Grammar  could  be  devised ; 
and  musical  compositions,  previous  to  the  sci- 

^    °f  ^^       ThlS'    b7  the  Way,  Serves  tO  6X- 

tiou  and     pose  the  futility  of  the  popular  objection  against 

classification 

of  principles.  -Logic ;  viz.  that  men  may  reason  very  well  who 
know  nothing  of  it.  The  parallel  instances  ad 
duced  show  that  such  an  objection  may  be  urged 


CHAP.   III.]  ANALYTICAL     OUTLINE.  61 

in  many  other  cases,  where  its  absurdity  would  Logic 
be  obvious  ;  and  that  there  is  no  ground  for  de 
ciding  thence,  either  that  the  system  has  no  ten 
dency  to  improve  practice,  or  that  even  if  it  had 
not,  it  might  not  still  be  a  dignified  and  inter 
esting  pursuit. 


§  43.    One  of  the  chief   impediments  to  the  sameness  of 

the  reasoning 

attainment  of  a  just  view  of  the  nature  and  ob-     process 
ject  of  Logic,  is  the  not  fully  understanding,  or 


not  sufficiently  keeping  in  mind  the   SAMENESS 
of  the  reasoning  process  in  all  cases.     If,  as  the 
ordinary  mode  of  speaking  would  seem  to  indi 
cate,   mathematical   reasoning,    and    theological,  AH  kinds  of 
and  metaphysical,  and  political,  &c.,  were  essen-  re<^lenfnare 
tially  different  from  each  other,  that  is,  different    PriuciPle- 
lands  of  reasoning,  it  would  follow,  that  suppo 
sing  there   could  be  at  all  any  such  science  as 
we  have  described  Logic,  there  must  be  so  many 
different  species   or  at  least   different   branches 
of  Logic.     And  such  is  perhaps  the  most  pre 
vailing  notion.     Nor  is   this  much  to  be  won-    Reason  of 

the  prevail- 

dered  at  ;  since  it  is  evident  to  all,  that  some    ing  error8> 
men  converse   and  write,  in  an  argumentative 
way,  very  justly  on  one  subject,  and  very  erro 
neously  on  another,  in  which  again  others  excel, 
who  fail  in  the  former. 

This  error  may  be  at  once  illustrated  and  re- 


62  LOGIC.  [BOOK  i. 


The  reason  of  moved,  by  considering  the  parallel  instance  of 
Arithmetic  ;  in  which  every  one  is  aware  that 


by  example,  faG  process  of  a  calculation  is  not   affected  bv 

which  shows  '  •> 

that  the  rea-  the  nature  of  the  objects  whose    numbers   are 

soning 

process  is     before  us  ;  but  that,   for  example,   the  multipli- 

always  the  .  c  -.  . 

same.  cation  oi  a  number  is  the  very  same  operation, 
whether  it  be  a  number  of  men,  of  miles,  or  of 
pounds  ;  though,  nevertheless,  persons  may  per 
haps  be  found  who  are  accurate  in  the  results 
of  their  calculations  relative  to  natural  philoso 
phy,  and  incorrect  in  those  of  political  econo 
my,  from  their  different  degrees  of  skill  in  the 
subjects  of  these  two  sciences  ;  not  surely  be 
cause  there  are  different  arts  of  arithmetic  ap 
plicable  to  each  of  these  respectively. 

§  44.    Others  again,  who  are  aware  that  the 

l°ogicVaST   simple  system  of  L°gic  ma7  be  applied  to  all 
peculiar     subjects  whatever,  are  yet  disposed  to  view  it 

method  of 

reasoning:  as  a  peculiar  method  of  reasoning,  and  not,  as 
it  is,  a  method  of  unfolding  and  analyzing  our 
reasoning  :  whence  many  have  been  led  to  talk 
of  comparing  Syllogistic  reasoning  with  Moral 
reasoning;  taking  it  for  granted  that  it  is  pos 
sible  to  reason  correctly  without  reasoning  logi- 

it  is  the  only  cally  ;  which  is,  in  fact,  as  great  a  blunder  as  if 

method  of 

reasoning    any  one  were  to  mistake  grammar  for  a  pecu- 

correctly  :      i  •         7 

liar  language,  and  to  suppose  it  possible  to  speak 


CHAP.   III.]  ANALYTICAL     OUTLINE.  63 


correctly  without  speaking  grammatically.  They 
have,  in  short,  considered  Logic  as  an  art  of  rea 
soning  ;  whereas  (so  far  as  it  is  an  art)  it  is  the 
art  of  reasoning;  the  logician's  object  being,  not  it  lays  down 

rules,  not 

to  lay  down  principles  by  which  one  may  reason,   which  may, 

but  which 

but  by  which  all  must  reason,  even  though  they     mustbe 
are  not  distinctly  aware  of  them  :  —  to  lay  down 
rules,   not  which  may  be  followed  with  advan 
tage,    but   which    cannot   possibly   be    departed 

from  in   sound  reasoning.     These  misapprehen-    Misappre 

hensions  and 
sions   and  objections  being  such  as  lie  on  the    objections 

very  threshold  of  the  subject,  it  would  have  been 
hardly  possible,  without  noticing  them,  to  con 
vey  any  just  notion  of  the  nature  and  design  of 
the  logical  system. 


§  45.    Supposing   it   then   to  have  been  per-  operation  of 

,,      reasoning 

ceived  that  the  operation  of  reasoning  is  in  all    Sh0uidbe 
cases   the  same,   the  analysis  of  that  operation    *"*!** 
could  not  fail  to  strike  the  mind  as  an  interesting 
matter  of  inquiry.     And  moreover,  since  (appa 
rent)  arguments,  which  are  unsound  and  incon 
clusive,  are  so  often  employed,  either  from  error  Because  such 

....  .  analysis  is 

or  design;    and  since  even  those  who  are  not  necessaryto 
misled  by  these  fallacies,  are  so  often  at  a  loss   furuish  the 
to  detect  and  expose   them  in  a  manner  satis 
factory  to  others,  or  even  to  themselves  ;  it  could 
not  but  appear  desirable  to  lay  down  some  gen- 


64  LOGIC.  [BOOK  i. 


rules  for  the  eral  rules  of  reasoning,  applicable  to   all  cases; 
detection  of  ,     wnicn  a  person  might  be  enabled  the  more 

error  and  the      * 

discovery  of  readily  and  clearly  to  state  the  grounds  of  his 
own  conviction,  or  pf  his  objection  to  the  argu 
ments  of  an  opponent;  instead  of  arguing  at 
random,  without  any  fixed  and  acknowledged 
principles  to  guide  his  procedure.  Such  rules 

such  rules    wouici  }ye  analogous  to  those  of  Arithmetic,  which 

are  analogous 

to  the  rules  of  obviate  the  tediousness  and  uncertainty  of  cal- 

Arithmetic. 

,  culations  in  the  head  ;  wherein,  after  much  labor, 
different  persons  might  arrive  at  different  results, 
without  any  of  them  being  able  distinctly  to 
point  out  the  error  of  the  rest.  A  system  of 
such  rules,  it  is  obvious,  must,  instead  of  deserv- 
They  bring  ing  to  be  called  the  art  of  wrangling,  be  more 

the  parties,  in  ,         ,  .        . 

argument,  to  Justty  characterized  as  the  "art  of  cutting  short 
sue<     wrangling,"  by  bringing  the   parties  to  issue  at 
once,  if  not  to  agreement;   and   thus  saving  a 
waste  of  ingenuity: 


Every  con-  §  46  jn  pursumg  the  supposed  investigation, 
it  will  be  found  that  in  all  deductive  processes 
every  conclusion  is  deduced,  in  reality,  from  two 

Premises,  other  propositions  (thence  called  Premises)  ;  for 
though  one  of  these  may  be,  and  commonly  is, 
suPPressed,  it  must  nevertheless  be  understood 

understood,  as  admitted  ;  as  may  easily  be  made  evident  by 
supposing  the  denial  of  the  suppressed  premiss, 


CHAP.   III.]  ANALYTICAL     OUTLINE.  65 

which  will  at  once  invalidate  the  argument.     For 
example ;  in  the  following  syllogism  : 

"  Whatever  exhibits  marks  of  design  had  an  intelligent  author: 
The  world  exhibits  marks  of  design  ; 
Therefore,  the  world  had  an  intelligent  author  :" 

if  any  one  from  perceiving  that  "  the  world  ex 
hibits  marks  of  design,"  infers  that  "it  must  have      andis 
had  an  intelligent  author,"  though  he  may  not  be  "JJj^J0 
aware  in  his  own  mind  of  the  existence  of  any  ment,  though 

one  may  not. 

other  premiss,  he  will  readily  understand,  if  it  be    be  aware 
denied  that  "  whatever  exhibits  marks  of  design 
must  have  had  an  intelligent  author,"  that  the 
affirmative   of  that  proposition  is  necessary  to 
the  validity  of  the  argument. 


§  47.  When  one  of  the  premises  is  suppressed 

a  syllogism 

(which  for  brevity's  sake  it  usually  is),  the  argu-     with  one 
ment  is  called  an  Enthymeme.     For  example  :       suppressed. 

"  The  world  exhibits  marks  of  design, 
Therefore  the  world  had  an  intelligent  author." 

is  an  Enthymeme.     And  it  may  be  worth  while 

to  remark,  that,  when  the  argument  is  in  this   objections 

made  to  the 

state,  the  objections  of  an  opponent  are  (or  rather 


assertion  or 


appear  to  be)  of  two  kinds,  viz.  either  objections 
to  the  assertion  itself,  or  objections  to  its  force      ment- 
as  an  argument.      For  example :   in  the   above    Example, 
instance,  an  atheist  may  be  conceived  either  de- 

5 


LOGIC.  [BOOK  i. 


nvino-  that  the  world  does  exhibit  marks  of  de- 

Both  prera-       J      » 

K*  must  be  g^^  or  denying  that  it  follows  from  thence  that 


it  had  an  intelligent  author.     Now  it  is  impor- 
8°und:      tant  to  keep  in  mind  that  the  only  difference  in 
the  two  cases  is,  that  in  the  one  the  expressed 
premiss  is  denied,  in  the  other  the  suppressed; 
and  when    for  the  force  as  an  argument  of  either  premiss 
both  are  true,  ^         ^g  on  faQ  other  premiss  :  if  both  be  admit- 

the  conclu-  i 

sion  follows.  te(^  t^e  conclusion  legitimately  connected  wTith 
them  cannot  be  denied. 

§  48.  It  is  evidently  immaterial  to  the  argu 
ment  whether  the  conclusion  be  placed  first  or 
Premiss     last ;    but  it  may  be  proper  to   remark,   that   a 
toecondu-  premiss  placed  after  its  conclusion  is  called  the 
sion  is  called  Reason  of  jt  and  [s  introduced  by  one  of  those 

the  Reason. 

conjunctions  which  are  called  causal,  viz.  "  since," 

"because,"  &c.,  which  may  indeed  be  employed 

to  designate  a  premiss,  whether  it  come  first  or 

niative     last.     The  illative  conjunctions  "  therefore,"  &c., 

llon'  designate  the  conclusion. 

It  is  a  circumstance   .which   often   occasions 
causes  of    error  anj  perplexity,  that  both  these  classes  of 

error  and 

perplexity,  conjunctions  have  also  another  signification,  be 
ing  employed  to  denote,  respectively,  Cause  and 
Effect,  as  well  as  Premiss  and  Conclusion.  For 

Different 

significations  example  i  if  I  say,  "  this  ground  is  rich,  because 

of  the 

conjunctions,  the  trees  on  it  are  flourishing ;"  or,  "  the  trees  are 


CHAP.   III.]  ANALYTICAL     OUTLINE.  67 

flourishing,  and  therefore  the  soil  must  be  rich  ;"    Examples 
I  employ  these  conjunctions  to  denote  the  con- 


nection  of  Premiss  and   Conclusion  ;    for  it  is     are  used 

logically. 

plain  that  the  luxuriance  of  the  trees  is  not  the 
cause  of  the  soil's  fertility,  but  only  the  cause 
of  my  knowing  it.     If  again  I  say,  "the  trees 
flourish,  because  the   ground  is  rich  ;"  or   "  the 
ground  is  rich,  and  therefore  the  trees  flourish/'     Examples 
I  am  using  the  very  same  conjunctions  to  denote  d^™  J.^e 
the  connection  of  cause  and  effect;  for  in  this    andeffect- 
case,  the  luxuriance  of  the  trees  being  evident 
to  the  eye,  would  hardly  need  to  be  proved,  but 
might  need  to  be   accounted  for.      There   are,  Many  cases 
however,  many  cases,  in  which  the  cause  is  em-  lnwhlc 


ployed  to  prove  the  existence  of  its  effect  :  es'pe-    the  reason 

are  the  same. 

cially  in  arguments  relating  to  future  events;  as, 
for  example,  when  from  favorable  weather  any 
one  argues  that  the  crops  are  likely  to  be  abun 
dant,  the  cause  and  the  reason,  in  that  case,  co 
incide  ;  and  this  contributes  to  their  being  so 
often  confounded  together  in  other  cases. 

§  49.    In  an  argument,   such  as  the  example     iu  every 
above  given,  it  is,  as  has  been  said,  impossible 


for  any  one,  who  admits  both  premises,  to  avoid    admit  the 

premiss  is  to 

admitting  the  conclusion.     But  there  will  be  fre-    admit  the 

conclusion. 

quently  an  apparent  connection  of  premises  with 
a   conclusion  which  does  not  in  reality  follow 


GS 


LOGIC.  [BOOK  r. 


Apparent  from  them,  though  to  the  inattentive  or  unskilful 
preT^and  the  argument  may  appear  to  be  valid ;  and  there 
conclusion  ^IQ  manv  other  cases  in  which  a  doubt  may  exist 

must  not  be 

relied  on.  whether  the  argument  be  valid  or  not ;  that  is, 
whether  it  be  possible  or  not  to  admit  the  prem 
ises  and  yet  deny  the  conclusion. 

General  rules      §  50.    It  is  of  the  highest  importance,  there- 

f°r  tetk>Ten"  *°re>  to  ^ay  down  some  regular  f°rm  to  which 
necessary.    everv  valid  argument  may  be  reduced,  and  to 

devise  a  rule  which  shall  show  the  validity  of 
every  argument  in  that  form,  and  consequently 
the  unsoundness  of  any  apparent  argument  which 
cannot  be  reduced  to  it.  For  example ;  if  such 
an  argument  as  this  be  proposed : 

Example  of  "  Every  rational  agent  is  accountable  : 

an  imperfect  Brutes  are  not  rational  agents  ; 

argument. 

Therefore  they  are  not  accountable  ;" 
or  again : 

2d  Example.    "  All  wise  legislators  suit  their  laws  to  the  genius  of  their 

nation  ; 
Solon  did  this ;  therefore  he  was  a  wise  legislator  :" 

Difficulty  of  there    are    some,  perhaps,  who  would  not  per- 

detecting  the         •  c  -\  i  •  i 

error  ceive  any  fallacy  in  such  arguments,  especially 
if  enveloped  in  a  cloud  of  words ;  and  still  more, 
wrhen  the  conclusion  is  true,  or  (which  comes  to 
the  same  point)  if  they  are  disposed  to  believe 
it ;  and  others  might  perceive  indeed,  but  might 


CHAP.  III.]  ANALYTICAL     OUTLINE.  69 

be  at  a  loss  to  explain,  the  fallacy.     Now  these     TO  what 
(apparent)    arguments    exactly   correspond,   re 
spectively,  with  the  following,  the  absurdity  of 
the  conclusions  from  which  is  manifest : 

"  Every  horse  is  an  animal :  A  similar 

Sheep  are  not  horses  ;  example" 

Therefore,  they  are  not  animals." 

And: 

"  All   vegetables  grow  ;  2d  similar 

An  animal  grows  ;  example. 

Therefore,  it  is  a  vegetable." 

These  last  examples,  I  have  said,  correspond    These  last 

.  \-iir  correspond 

exactly  (considered  as  arguments)  with  the  for-     with  the 
mer ;  the  question  respecting  the  validity  of  an     formcr- 
argument  being,  not  whether  the  conclusion  be 
true,  but  whether  it  follows  from  the  premises 
adduced.     This  mode  of  exposing  a  fallacy,  by 
bringing  forward  a  similar  one  whose  conclusion 
is  obviously  absurd,  is  often,  and  very  ad  van-      times 

J  -  resorted  to. 

tageously,  resorted  to  in  addressing  those  who 
are  ignorant  of  Logical  rules ;  but  to  lay  down 
such  rules,  and  employ  them  as  a  test,  is  evi- 


t.  T  n     rules  is  the 

dently  a   sater  and  more   compendious,  as  well    best  way. 
as  a  more  philosophical  mode  of  proceeding.     To 
attain  these,   it  would  plainly  be    necessary  to 
analyze  some  clear  and  valid  arguments,  and  to 
observe  in  what  their  conclusiveness  consists. 


70  LOGIC.  [BOOK  i. 

§  51.  Let  us  suppose,  then,  such  an  examin 
ation  to  be  made  of  the  syllogism  above  men 
tioned  : 

Example  of  "  Whatever  exhibits  marks  of  design  had  an  intelligent  author; 
a  perfect        The  worl(j  exhibits  marks  of  design  ; 

Therefore,  the  world  had  an  intelligent  author." 

what  is         In  the  first  of  these  premises  we  find  it  as- 

the  first     sumed  universally  of  the  class  of  "  things  which 

premiss.     exhibit  marks  of  design,"  that  they  had  an  intel- 

rn  the  second  ligent  author ;   and  in  the  other  premiss,  "the 

world"  is  referred  to  that  class  as  comprehended 

what  we    jn  ^  .  now  fa  js  evident  that  whatever  is  said  of 

may  infer. 

the  whole  of  a  class,  may  be  said  of  any  thing 
comprehended  in  that  class ;  so  that  we  are  thus 
authorized  to  say  of  the  world,  that  "  it  had  an 
intelligent  author." 
syllogism          Again,  if  we   examine    a    syllogism   with   a 

with  a 

negative     negative  conclusion,  as,  for  example, 

conclusion. 

"  Nothing  which  exhibits  marks  of  design  could  have  been 

produced  by  chance  ; 
The  world  exhibits,  &c.  ; 

Therefore,  the  world  could  not  have  been   produced  by 
chance," 

The  process  the  process  of  reasoning  will  be  found  to  be  the 

of  reasoning 

the  same,  same;  since  it  is  evident  that  whatever  is  denied 
universally  of  any  class  may  be  denied  of  any 
thing  that  is  comprehended  in  that  class. 


CHAP.   III.]  ANALYTICAL     OUTLINE.  71 


§  52.  On  further  examination,  it  will  be  found     AH  valid 
that  all   valid  arguments  whatever,  which   are  reducible  to 
based  on  admitted  premises,  may  be  easily  re- 
duced  to  such  a  form  as  that  of  the  foregoing 
syllogisms ;  and  that  consequently  the  principle 
on  which  they  are  constructed  is  that  of  the  for 
mula  of  the  syllogism.    So  elliptical,  indeed,  is  the 
ordinary  mode  of  expression,  even  of  those  who 
are  considered  as  prolix  writers,  that  is,  so  much   expressing 

arguments 

is  implied  and  left  to  be  understood  in  the  course  elliptical. 
of  argument,  in  comparison  of  what  is  actually 
stated  (most  men  being  impatient  even,  to  excess, 
of  any  appearance  of  unnecessary  and  tedious 
formality  of  statement),  that  a  single  sentence 
will  often  be  found,  though  perhaps  considered 
as  a  single  argument,  to  contain,  compressed 
into  a  short  compass,  a  chain  of  several  distinct 
arguments.  But  if  each  of  these  be  fully  devel-  Butwhen 

fully  devel 
oped,  and  the  whole  of  what  the  author  intended   oped,  they 

may  all  be 

to  imply  be  stated  expressly,  it  will  be  found  that  reduced  into 
all  the  steps,  even  of  the  longest  and  most  com 
plex  train  of  reasoning,  may  be  reduced  into  the 
above  form. 


§  53.  It  is  a  mistake  to  imagine  that  Aristotle 
and  other  logicians  meant  to  propose  that  this 
prolix  form  of  unfolding  arguments  should  uni-  that  every 

argument 

versally  supersede,  in  argumentative  discourses,    should  i>e 


the  above 
form. 


72  LOGIC.  [BOOK  i. 


thrown  into  the  common  forms  of  expression ;  and  that  "  to 

ie  form  of 

syllogism. 


a  reason  logically,"  means,  to  state  all  arguments 


at  full  length  in  the  syllogistic  form ;  and  Aris 
totle  has  even  been  charged  with  inconsistency 
for  not  doing  so.     It  has  been  said  that  he  "  ar 
gues  like  a  rational  creature,  and  never  attempts 
That  form  is  to  bring  his  own  system  into  practice."     As  well 
"on'ruth***  might  a  chemist  be  charged  with  inconsistency 
for  making  use  of  any  of  the    compound   sub 
stances  that  are    commonly  employed,   without 
previously    analyzing    and   resolving   them    into 
Analogy  to   their  simple  elements;  as  well  might  it  be  im- 

the  chemist. 

agined  that,  to  speak  grammatically,  means,  to 
parse  every  sentence  we  utter.  The  chemist 
(to  pursue  the  illustration)  keeps  by  him  his  tests 
and  his  method  of  analysis,  to  be  employed  when 
The  analogy  any  substance  is  offered  to  his  notice,  the  com- 

uuntiuued. 

position  of  which  has  not  been  ascertained,  or 

in  which  adulteration  is  suspected.     Now  a  fal- 

Towhata    lacy  may  aptly  be  compared  to  some  adulterated 

fallacy  may 

be  compared,  compound ;  "it  consists  of  an  ingenious  mixture 
of  truth  and  falsehood,  so  entangled,  so  intimate 
ly  blended,  that  the  falsehood  is  (in  the  chemical 
phrase)  held  in  solution  :  one  drop  of  sound  logic 

low  detect-  is  tllat  test  which  immediately  disunites  them, 
makes  the  foreign  substance  visible,  and  precipi 
tates  it  to  the  bottom." 


CHAP.  III.]  ANALYTICAL     OUTLINE.  73 


ARISTOTLES    DICTUM. 

§  54.  But  to  resume  the  investigation  of  the     Form  of 

every  reaJ 
argument. 


principles  of  reasoning :  the  maxim  resulting  from 


the  examination  of  a  syllogism  in  the  foregoing 
form,  and  of  the  application  of  which,  every  valid 
deduction  is  in  reality  an  instance,  is  this  : 

"  That  whatever  is  predicated  (that  is,  affirmed   Aristotle's 
or  denied)  universally,  of  any  class  of  things, 
may  be  predicated,  in  like  manner  (viz.  affirmed 
or  denied),  of  any  thing  comprehended  in  that 
class." 

This  is  the  principle  commonly  called  the  die-    what  the 
turn   de    omni    et   nullo,    for    the    indication   of 
which  we  are  indebted  to  Aristotle,  and  which 
is  the  keystone  of  his  whole  logical  system.     It 
is    remarkable    that    some,    otherwise   judicious  wh.^^^ 
writers,   should  have  been  so  carried   away  by  havesaidof 

this  princi- 

their  zeal  against  that  philosopher,  as  to  speak    pie;  and 

why. 

with   scorn    and   ridicule    of   this    principle,    on 
account    of     its    obviousness    and    simplicity  ;  simplicitya 
though  they  would   probably  perceive  at  once      testof 
in   any  other  case,  that   it   is   the   greatest   tri 
umph  of  philosophy   to  refer  many,  and   seem 
ingly  very  various  phenomena  to  one,  or  a  very 
few,  simple  principles ;  and  that  the  more  simple 
and  evident  such  a  principle  is,  provided  it  be 
truly  applicable  to  all  the  cases  in  question,  the 


74  LOGIC.  [BOOK  i. 

NO  solid  ob-  greater  is  its  value  and  scientific  beauty.      If, 
'^priTcipte116  indeed,  any  principle  be  regarded  as  not  thus  ap- 
ever  urged.   piicable,  that  is  an  objection  to  it  of  a  different 
kind.     Such  an  objection  against  Aristotle's  dic 
tum,  no  one  has  ever  attempted  to  establish  by 
blntakra   an7  kind  °f  proof ;  but  it  has  often  been  taken 
for  granted,  fa  grantea ;  it  being  (as  has  been  stated)  very 
syllogism    commonly  supposed,  without    examination,  that 

not  a  distinct 

kind  of  ar-   tne  syllogism  is  a  distinct  kind  of  argument,  and 


;       that  the  rules  of  it  accordingly  do  not  apply,  nor 
applicable  to  were  intended  to  apply,  to  all  reasoning  what- 

all  cases. 

ever,  where  the  premises  are  granted  or  known. 

objection:        §  55.  One  objection  against  the  dictum  of  Aris 
totle  it  may  be  worth  while  to  notice  briefly,  for 


intended  to  tne  sake  of  setting  in  a  clearer  light  the   real 

make  a  dem 
onstration    character  and  object  of  that  principle.     The  ap- 

plainer : 

plication  of  the  principle  being,  as  has  been 
seen,  to  a  regular  and  conclusive  syllogism,  it 
has  been  urged  that  the  dictum  was  intended 
to  prove  and  make  evident  the  conclusiveness 
of  such  a  syllogism;  and  that  it  is  unphilo- 
sophical  to  attempt  giving  a  demonstration  of 
a  demonstration.  And  certainly  the  charge 
to  increase  would  be  just,  if  we  could  imagine  the  logi- 

the  certainty 

of  a  cian  s  object  to  be,  to  increase  the  certainty 
of  a  conclusion,  which  we  are  supposed  to  have 
already  arrived  at  by  the  clearest  possible  mode 


CHAP.   III.]  ANALYTICAL     OUTLINE.  75 

of  proof.     But  it  is  very  strange  that  such  an  This  view  is 
idea  should  ever  have  occurred  to  one  who  had   erroneous. 
even  the  slightest  tincture  of  natural  philosophy ; 
for  it  might  as  well  be  imagined  that  a  natural  illustration, 
philosopher's  or  a  chemist's  design  is  to  strength 
en  the  testimony  of  our  senses  by  a  priori  rea 
soning,  and  to  convince  us  that  a  stone  when 
thrown  will  fall  to  the  ground,  and  that  gunpow 
der  will  explode  when  fired ;  because  they  show 
according  to  their  principles  those   phenomena 
must  take  place  as  they  do.     But  it  would  be 
reckoned  a  mark  of  the  grossest  ignorance  and 

.  The  object  is 

stupidity  not  to  be  aware  that  their   object   is  not  to  prove, 


not  to  prove  the  existence  of  an  individual 
phenomenon,  which  our  eyes  have  witnessed, 
but  (as  the  phrase  is)  to  account  for  it ;  that  is, 
to  show  according  to  what  principle  it  takes 
place ;  to  refer,  in  short,  the  individual  case  to 
a  general  law  of  nature.  The  object  of  Aris-  ^object of 

J  the  Dictum 

totle's  dictum  is  precisely  analogous:    he    had,   to  point  out 

the  general 

doubtless,  no  thought  of  adding  to  the  force  of  process  to 

T  •     •  i       i         IT-  i  •       i  •    j.     which  each 

any  individual  syllogism  ;  his  design  was  to  point     case  con. 
out  the  general  principle  on  which  that  process 
is  conducted  which  takes  place  in  each   syllo 
gism.     And  as  the  Laws  of  nature  (as  they  are     Laws  of 

nature,  gen- 

c ailed)  are  in  reality  merely  generalized  facts,  of  erased  facts, 
which  all  the  phenomena  coming  under  them  are 
particular  instances ;  so,  the  proof  drawn  from 


76  LOGIC.  [BOOK  i. 


The  Dictum  Aristotle's  dictum  is  not  a  distinct  demonstration 

af™"djfn^,d  brought  to  confirm  another  demonstration,  but  is 

demonstra-   mereiy  a  generalized  and  abstract  statement  of 

all  demonstration  whatever ;  and  is,  therefore,  in 

fact,  the  very  demonstration  which,  under  proper 

suppositions,  accommodates  itself  to  the  various 

subject-matters,  and  which  is  actually  employed 

in  each  particular  case. 

FIOW  to  trace      §56.    In  order  to  trace  more  distinctly  the 

the  abstract-  ~  , 

ing  and     different   steps   of   the    abstracting   process,    by 


any  particular  argument  may  be  brought 
into  the  most  general  form,  we  may  first  take  a 
syllogism,  that  is,  an  argument  stated  accurately 
AD  argument  arid  at  full  length,  such  as  the  example  formerly 

stated  at  full 

length,      given : 

"  Whatever  exhibits  marks  of  design  had  an  intelligent  author; 
The  world  exhibits  marks  of  design ; 
Therefore,  the  world  had  an  intelligent  author :" 

Propositions  and  then  somewhat  generalize  the  expression,  by 

ex^eg^dtby  substituting   (as  in  Algebra)  arbitrary  unmean- 

terms.      ing  Symbols  for  the  significant  terms  that  were 

originally  used.     The  syllogism  will  then  stand 

thus  : 

"  Every  B  is  A ;  C  is  B  ;  therefore  C  is  A." 
The  reason-       The  reasoning,  when  thus  stated,  is  no  less  evi- 

ing  no  less 

valid,      dently  valid,  whatever  terms  A,  B,  and  C  respect- 


CHAP.   III.]  ANALYTICAL     OUTLINE.  77 


ively  may  be  supposed  to  stand  for ;  such  terms       and 
may  indeed  be  inserted  as  to  make  all  or  some     general, 
of  the  assertions  false ;  but  it  will  still  be  no  less 
impossible  for  any  one  who  admits  the  truth  of 
the  premises,  in  an  argument  thus  constructed, 
to  deny  the  conclusion ;  and  this  it  is  that  con 
stitutes  the  conclusiveness  of  an  argument. 

Viewing,  then,   the  syllogism  thus  expressed,  syiiogismso 

viewed, 

it  appears  clearly  that  "  A  stands  for  any  thing  affirms  gen- 
whatever  that  is  affirmed  of  a  certain  entire  class"  ^tJTen  the* 
(viz.  of  every  B),  "which  class  comprehends  or 
contains  in  it  something  else;"  viz.  C  (of  which  B 
is,  in  the  second   premiss,  affirmed)  ;    and  that, 
consequently,  the  first  term  (A)  is,  in  the  conclu 
sion,  predicated  of  the  third  (C). 


§  57.  Now,  to  assert  the  validity  of  this  pro-  Another  form 

of  stating  the 

cess  now  before  us,  is  to  state  the  very  dictum     dictum> 
we  are  treating  of,  with  hardly  even  a  verbal 
alteration,  viz.  : 

1.  Any  thing  whatever,  predicated  of  a  whole    The  three 

things 
ClaSS  ;  implied. 

2.  Under  which  class  something  else  is  con 
tained  ; 

3.  May  be  predicated  of  that  which  is  so  con- 

,  These  three 

tamed-  members 

The  three  members  into  which  the  maxim  is  correspond  to 

the  three 

here  distributed,  correspond  to  the  three  propo-  propositions 


78  LOGIC. 


[BOOK  i. 


sitions  of  the  syllogism  to  which  they  are   in 
tended  respectively  to  apply. 

Advantage  of      The  advantage  of  substituting  for  the  terms, 
in  a  regular  syllogism,  arbitrary,  unmeaning  sym- 
ls,  suc}1  as  letters  of  the  alphabet,  is  much  the 


the  terms. 

same  as  in  geometry  :  the  reasoning  itself  is  then 
considered,  by  itself,  clearly,  and  without  any 
risk  of  our  being  misled  by  the  truth  or  falsity 
of  the  conclusion  ;  which  is,  in  fact,  accidental 
and  variable;  the  essential  .point  being,  as  far  as 
connection,  faQ  argument  is  concerned,  the  connection  be- 

the  essential 

point  of  the  tween  the  premises  and  the  conclusion.     We  are 

argument. 

thus  enabled  to  embrace  the  general  principle  of 
deductive  reasoning,  and  to  perceive  its  applica 
bility  to  an  indefinite  number  of  individual  cases. 
That  Aristotle,  therefore,  should  have  been  ac- 
Aristotie  cuseci  of  making  use  of  these  symbols  for  the 

right  in  using 

these  sym-    purpose    of  darkening   his   demonstrations,   and 

bols. 

that  too  by  persons  not  unacquainted  with  geom 
etry  and  algebra,  is  truly  astonishing. 


syllogism        §  5g.  It  belongs,  then,  exclusively  to  a  syilo- 

cqually  true 

whenab-    gism,  properly  so  called  (that  is,  a  valid  argu- 

Btract  terms  ,      , 

are  used.  ment,  so  stated  that  its  conclusiveness  is  evident 
from  the  mere  form  of  the  expression),  that  if 
letters,  or  any  other  unmeaning  symbols,  be  sub 
stituted  for  the  several  terms,  the  validity  of  the 
argument  shall  still  be  evident.  Whenever  this 


CHAP.   III.]  ANALYTICAL     OUTLINE.  79 

is  not  the  case,  the  supposed  argument  is  either  whennotso, 

....  ,   the  supposed 

unsound  and  sophistical,  or  else  may  be  reduced    argument 
(without  any  alteration  of  its  meaning)  into  the  li 
syllogistic  form ;    in  which  form,   the  test  just 
mentioned  may  be  applied  to  it. 

§  59.  What  is  called  an  unsound  or  fallacious  Definition  of 

an  unsound 

argument,  that  is,  an  apparent  argument,  which    argument, 
is,  in  reality,  none,  cannot,  of  course,  be  reduced 
into  this  form ;  but  when  stated  in  the  form  most 

nearly  approaching  to  this   that  is  possible,  its    when  re 
duced  to  the 
fallaciousness  becomes   more   evident,   from   its  form,  the  fai- 

nonconformity  to  the  foregoing  rule.     For  ex-  ^1™^°™ 
ample  : 

"  Whoever  is  capable  of  deliberate  crime  is  responsible  ;          Example. 
An  infant  is  not  capable  of  deliberate  crime  ; 
Therefore,  an  infant  is  not  responsible." 

Here  the  term  "responsible"  is  affirmed  uni-    Analysis  of 

•  c  •  ))  this  syllogism. 

versally  of  "  those  capable  of  deliberate  crime ; 
it  might,  therefore,  according  to  Aristotle's '  dic 
tum,  have  been  affirmed  of  any  thing  contained 
under  that  class ;  but,  in  the  instance  before  us, 
nothing  is  mentioned   as  contained  under  that  its  defective 
class ;  only,  the  term  "  infant"  is  excluded  from 
that  class;    and   though  what  is  affirmed  of  a 
whole  class  may  be  affirmed  of  any  thing  that 
is  contained  under  it,  there  is  no  ground  for  sup 
posing  that  it  may  be  denied  of  whatever  is  not 


80  LOGIC.  [BOOK  r. 


so  contained ;  for  it  is  evidently  possible  that  it 
the  lament  ma7  be  applicable  to  a  whole  class  and  to  some- 
is  not  good.  tnmg  ejge  besides.     To  say,  for  example,  that  all 
trees  are  vegetables,  does  not  imply  that  nothing 
else  is  a  vegetable.     Nor,  when  it  is  said,  that 

what  the    ajj  wno  are  capable  of  deliberate  crime  are  re 
statement 
implies,     sponsible,  does   this   imply  that   no   others    are 

responsible;  for  though  this  may  be  very  true, 
what  is  to   it  has  not  been  asserted  in  the  premiss  before  us ; 

be  done  in  . 

the  analysis  and  in  the  analysis  of  an  argument,  we  are  to 
argument     discard  all  consideration  of  what  might  be  as 
serted  ;  contemplating  only  what  actually  is  laid 
down  in  the  premises.     It  is  evident,  therefore, 
The  one     that  such  an   apparent   argument  as  the   above 
comply  with  does  n°t  comply  with  the  rule  laid  down,  nor 
the  rule.     can  be  so  stated  as  to  comply  with  it,  and  is 
consequently  invalid. 


§  60.  Again,  in  this  instance  : 

Another  "  Food  is  necessary  to  life  ; 

example.  Corn  is  food  ; 

Therefore  corn  is  necessary  to  life  :" 

in  what  the  tne  term  "necessary  to  life"  is  affirmed  of  food, 

argument  ia 

defective,  but  not  universally ;  for  it  is  not  said  of  every 
kind  of  food:  the  meaning  of  the  assertion  be 
ing  manifestly  that  some  food  is  necessary  to 
life :  here  again,  therefore,  the  rule  has  not  been 
complied  with,  since  that  which  has  been  predi- 


CHAP.   III.]  ANALYTICAL     OUTLINE.  81 

cated   (that  is,   affirmed  or  denied),   not  of  the     why  we 
whole,  but  of  a,  part  only  of  a  certain  class,  can- 


not  be,  on  that  ground,  predicated  of  whatever    whatwas 

predicated  of 

is  contained  under  that  class.  lood- 


DISTRIBUTION  AND  NON-DISTRIBUTION  OF  TERMS. 

§  61.  The  fallacy  in  this  last  case  is,  what  is  Fallacy  in  the 

last  example. 

usually  described  in  logical  language  as  consist 
ing  in  the  "  non-distribution  of  the  middle  term  ;"  Non-distribu 
tion  of  the 
that  is,  its  not  being  employed  to  denote  all  the  middle  term. 

objects  to  which  it  is  applicable.  In  order  to 
understand  this  phrase,  it  is  necessary  to  observe, 
that  a  term  is  said  to  be  "  distributed,"  when  it  is 
taken  universally,  that  is,  so  as  to  stand  for  all 
its  significates ;  and  consequently  "  undistribu 
ted,"  when  it  stands  for  only  a  portion  of  its  sig 
nificates.*  Thus,  "all  food,"  or  every  kind  of  what dtstn- 

.  ,          ,  ,.        .,          *uf  Mm  means. 

food,  are  expressions  which  imply  the  distribu 
tion  of  the   term   "  food ;"    "  some   food"  would  Non-distribu 
tion, 
imply  its  non-distribution. 

Now,  it  is  plain,  that  if  in  each  premiss  a  part 
only  of  the  middle  term  is  employed,  that  is,  if 
it  be  not  at  all  distributed,  no  conclusion  can 

How  the  ex- 
be  drawn.     Hence,  if  in  the  example  formerly  ample  might 

adduced,  it  had  been  merely  stated  that  "  some-      varied " 


*  Section  15. 
6 


82  LOGIC.  [BOOK  i. 

thing"    (not    "  whatever''    or    "  every    thing"} 

"  which  exhibits  marks  of  design,  is  the  work  of 

an   intelligent    author,"    it  would  not  have   fol- 

whatit     lowe(i  from  the  world's  exhibiting  marks  of  de- 

wmikt  then  ° 

sign,  that  that  is. the  work  of  an  intelligent  author. 


words  mark-      §  62.  It  is  to  be  observed  also,  that  the  words 
all"  and  "  every,"  which  mark  the  distribution 


distribution   Q£  a  te        and  «some»'  which  marks  its  non- 

not  always 

expressed,  distribution,  are  not  always  expressed :  they  are 
frequently  understood,  and  left  to  be  supplied  by 
the  context;  as,  for  example,  "food  is  neces 
sary  ;"  viz.  "  some  food  ;"  "  man  is  mortal ;"  viz. 
suchpropo-  "every  man."  Propositions  thus  expressed  are 
called  called  by  logicians  "  indefinite,"  because  it  is  left 
undetermined  by  the  form  of  the  expression 
whether  the  subject  be  distributed  or  not.  Nev 
ertheless  it  is  plain  that  in  every  proposition 
the  subject  either  is  or  is  not  meant  to  be  dis 
tributed,  though  it  be  not  declared  whether 

But  every    it  is  or    not ;    consequently,   every   proposition, 

proposition  ,.-,., 

must  be     whether  expressed   indefinitely  or  not,  must  be 
either      understood    as  either  "universal"  or    "particu- 

Umversal  or 

Particular.  Jar '"  those  being  called  universal,  in  which  the 
predicate  is  said  of  the  whole  of  the  subject 
(or,  in  other  words,  where  all  the  significates 
are  included) ;  and  those  particular,  in  which 

Example  of 

each.       only  a  part  of  them  is  included.     For  example  : 


CHAP.   III.]  ANALYTICAL     OUTLINE.  83 


"All  men  are  sinful/'  is  universal:   "some  men  This  division 


relates  to 


are  sinful,"  particular ;  and  this  division  of  prop- 

*  quantity. 

ositions,  having  reference  to  the  distribution  of 
the  subject,  is,  in  logical  language,  said  to  be  ac 
cording  to  their  "  quantity." 


§  63.  But  the  distribution  or  non-distribution  Distribution 

of  the  predi- 

of  the  predicate  is  entirely  independent  of  the 
quantity  of  the  proposition  ;  nor  are  the  signs 
"  all"  and  "  some"  ever  affixed  to  the  predicate  ; 
because  its  distribution  depends  upon,  arid  is  Has  reference 

to  quality  . 

indicated  by,  the  "  quality"  of  the  proposition  ; 
that  is,  its  being  affirmative  or  negative  ;  it  being 
a  universal  rule,  that  the  predicate  of  a  negative 
proposition  is  distributed,  and  of  an  affirmative,  Whonitis 

distributed: 

undistributed.     The  reason    of  this   may  easily 

be  understood,  by  considering  that  a  term  which    The  reason 

J  of  this. 

stands  for  a  whole  class  may  be  applied  to  (that 
is,  affirmed  of)  any  thing  that  is  comprehended 
under  that  class,  though  the  term  of  which  it  is 
thus  affirmed  may  be  of  much  narrower  extent 
than  that  other,  and  may  therefore  be  far  from  may  be  aP' 

plicable  to 

coinciding  with  the  whole  of  it.     Thus  it  may  the  subject. 

•1-1  1  TVT  •  aiU'   J'Ct   °f 

be  said  with  truth,  that  "the  Negroes  are  unciv-  much  wider 
ilized,"  though  the  term  "  uncivilized"  be  of  much 
wider  extent  than   "Negroes,"    comprehending, 
besides    them,    Patagonians,    Esquimaux,    &c.  ; 


so  that  it  would  not  be  allowable  to  assert,  that 


84  LOGIC.  [BOOK  i. 

Hence,  oniya  all  who  are  uncivilized  are  Negroes."     It  is  ev- 

tt™  ifu^d.  Went,    therefore,    that  it  is   a  part  only  of  the 

term    "uncivilized"  that    has  been    affirmed  of 

"  Negroes  ;"    and  the  same  reasoning  applies  to 

every  affirmative  proposition. 

But  it  may       It   may  indeed   so   happen,   that   the   subject 

Mtei^wuh   and   predicate    coincide,   that   is,    are    of  equal 

the  subject:  extent  j  as,  for  example:  "all  men  are  rational 

animals  ;"  "  all  equilateral  triangles  are  equian 

gular  ;"  (it  being  equally  true,  that  "  all  rational 

this  not  im-   animals  are  men,"  and  that  "all  equiangular  tri 

form  of  the    angles  are  equilateral  ;")  yet  this  is  not  implied 

llon'    by  the  form  of  the  expression  ;    since  it  would 

be  no  less  true  that  "all  men  are  rational  ani 

mals,"  even  if  there  were  other  rational  animals 

besides  men. 

if  any  part  of      It  is  plain,  therefore,  that  if  any  part  of  the 

the  predicate  ,.  .  ,.       ,  ,  ,  .  .  , 

is  applicable  predicate  is  applicable  to  the  subject,  it  may  be 
to  the  sub-    affirmed,  and  Of  course  cannot  be  denied,  of  that 

ject,  it  may 

be  affirmed    subject  ;   and  consequently,  when  the   predicate 

of  the  sub-  J 

ject.        is    denied   of  the  subject,   this  implies   that    nt, 

part  of  that  predicate  is  applicable  to  that  sub 

ject  ;  that  is,  that  the  whole  of  the  predicate  is 

if  a  predicate  denied  of  the  subject  :  for  to  say,  for  example, 

is  denied,  of  a     , 

subject,  the  tnat  "  no  beasts  of  prey  ruminate,    implies  that 


Beasts  °f  Prev  are  excluded  from  the  whole  class 
the  suvet    °^  ruminant  am'mals>  and  consequently  that  "  no 
ruminant  animals  are   beasts    of    prey."      And 


CHAP.  III.]  ANALYTICAL     OUTLINE.  SL 


hence  results  the  above-mentioned  rule,  that  the  Distribution 
distribution  of  the  predicate  is  implied  in  nega- 
tive  propositions,  and  its  non-distribution  in  af- 

propositions : 

firmativeS.  non-distribu 

tion  in 
affirmatives. 

§  64.  It  is  to  be  remembered,  therefore,  that  Not  sufficient 

for  the  mid- 
it  is  not  sufficient  for  the  middle  term  to  occur  die  term  to 

.  ,  .   .  .  .,,     ,  occur  in  a 

in  a  universal  proposition ;  since  11  that  propo-    universai 

sition  be  an  affirmative,  and  the  middle  term  be  Pr°P°sition- 

the  predicate  of   it,  it  will   not  be   distributed. 

For  example :  if  in  the  example  formerly  given, 

it  had  been  merely  asserted,  that  "  all  the  works 

of  an  intelligent  author  show  marks  of  design," 

and  that  "the  universe  shows  marks  of  design," 

nothing  could  have  been  proved ;  since,  though 

both  these  propositions  are  universal,  the  middle  terms  of  the 

conclusion, 

term  is  made  the  predicate  in  each,  and  both  are    that  those 

terms  may  be 

affirmative;  and  accordingly,  the  rule  of  Aris-  compared to- 
totle  is  not  here  complied  with,  since  the  term 
"  work  of  an  intelligent  author,"  which  is  to  be 
proved  applicable  to  "  the  universe,"  would  not 
have  been  affirmed  of  the  middle  term  ("  what 
shows  marks  of  design")  under  which  "  universe" 
is  contained ;  but  the  middle  term,  on  the  con 
trary,  would  have  been  affirmed  of  it. 

If,  however,  one  of  the  premises  be  negative,  if  One  prem- 
the  middle  term  may  then  be  made  the  predicate 


8(5  LOGIC.  [BOOK  i. 


uve,  the  mid-  of  that,   and  will  thus,   according  to  the   above 
bemwtetbT  remark,  be  distributed.     For  example  : 

predicate  of 

"be^dtetriiT11  "  ^°  rum'nant  anima^s  are  predacious  : 

uted.  The  lion  is  predacious  ; 

Therefore  the  lion  is  not  ruminant :" 

this  is  a  valid  syllogism  ;  and   the  middle  term 

(predacious)   is  distributed  by  being  made    the 

The  form  of  predicate  of  a  negative  proposition.     The  form, 


m(jeej    Of  the  syllogism  is  not   that  prescribed 

gism  will  not  J 

bethatpre-  by  the  dictum  of  Aristotle,  but  it  may  easily  be 

scribed  by 

the  dictum,   reduced  to  that  form,  by  stating  the  first  prop- 
but  may  be          .   .  1VT  .  .        1 

reduced  to  it.  osition  thus :  "IMo  predacious  animals  are  ru 
minant;"  which  is  manifestly  implied  (as  was 
above  remarked)  in  the  assertion  that  "  no  ru 
minant  animals  are  predacious."  The  syllogism 
will  thus  appear  in  the  form  to  which  the  dictum 
applies. 


Aiiargu-         §  65.  It  is  not  every  argument,   indeed,   that 

merits  cannot  . 

be  reduced   can  be  reduced  to  this  form  by  so  short  and  sim- 
lort  a  pie  an  alteration  as  in  the  case  before  us.     A 

process.       *• 

longer  and  more  complex  process  will  often  be 

required,  and  rules  may  be  laid  down  to  facilitate 

this  process  in  certain  cases ;    but  there  is  no 

sound  argument  but  what  can  be  reduced  into 

But  mi  argu-  this  form,  without  at  all  departing  from  the  real 

may   meaning  and  drift  of  it ;   and  the  form  will  be 


CHAP.   III.]  ANALYTICAL      OUTLINE.  87 


found   (though  more  prolix  than  is  needed  for   be  reduced 
ordinary  use)  the  most  per; 
argument  can  be  exhibited. 


v      ,  .  •  i  •    i  to  the  pre- 

ordmary  use)  the  most  perspicuous  in  which  an  scribed  form 


§  66.  All  deductive  reasoning  whatever,  then,  AH  deductive 

•        •    -I       i    •  i      i  ,  reasoning 

rests  on  the  one  simple  principle  laid  down  by  restSOnthe 
Aristotle,  that  dictura- 

"  What  is  predicated,  either  affirmatively  or 
negatively,  of  a  term  distributed,  may  be  predi 
cated  in  like  manner  (that  is,  affirmatively  or  neg 
atively)  of  any  thing  contained  under  that  term." 

So  that,  when  our  object  is  to  prove  any  prop-  what  are  the 
osition,  that  is,  to  show  that  one  term  may  rightly  J 
be  affirmed  or  denied   of  another,   the  process 
which  really  takes  place  in  our  minds  is,  that  we 
refer  that  term  (of  which  the  other  is  to  be  thus 
predicated)  to  some  class  (that  is,  middle  term) 
of  which  that  other  may  be  affirmed,  or  denied, 

as  the  case  may  be.     Whatever  the  subject-mat-   The  reason 
ing  always 

ter  of  an  argument  may  be,  the  reasoning  itself,     the  same, 
considered  by  itself,  is  in  every  case  the  same 
process;    and  if  the  writers  against  Logic  had   Mistakes  of 

writers  on 

kept  this  in  mind,  they  would  have  been  cautious      Logic. 
of  expressing  their  contempt  of  what  they  call 
"syllogistic  reasoning,"  which  embraces  all  de 
ductive  reasoning ;  and  instead  of  ridiculing  Aris 
totle's  principle  for  its  obviousness  and  simplicity,    Aristotle's 
would  have  perceived  that  these  are,  in  fact,  its 


88  LOGIC.  [BOOK  i. 


simple  and    highest  praise  i    the  easiest,  shortest,  and  most 
evident  theory,  provided  it  answe 
of  explanation,  being  ever  the  best. 


evident  theory,  provided  it  answer  the  purpose 


RULES    FOR    EXAMINING   SYLLOGISMS. 


Tests  of  the       §  67.  The  following  axioms  or  canons  serve 

validity  of 

syllogisms,    as   tests  of  the  validity  of  that  class  of  syllo 
gisms  which  we  have  considered. 
1st  test.          1st.  If  two  terms  agree  with  one  and  the  same. 

third,  they  agree  with  each  other. 

Sdtest.          2d.  If  one  term  agrees  and  another  disagrees 
with  one  and  the  same  third,  these  two  disagree 
with  each  other. 
The  first  the       On  the  former  of  these  canons  rests  the  va- 

t*.a*    ,-P  nil 


test  of  all 
affirmative 


lidity  of  affirmative  conclusions  ;  on  the  latter, 

conclusions. 


;onclusions.       r 

The  second  ot  negative:  tor,  no  syllogism  can  be  faulty 
which  does  not  violate  these  canons ;  none  cor 
rect  which  does;  hence,  on  these  two  canons 
are  built  the  following  rules  or  cautions,  which 
are  to  be  observed  with  respect  to  syllogisms, 
for  the  purpose  of  ascertaining  whether  those 
canons  have  been  strictly  observed  or  not. 

Every  syiio-       1st.  Every  syllogism  has  three  and  only  three. 

gismhas  . 

three  and    terms  >'  viz.  the  middle  term  and  the  two  terms 
0ntennTe    of  the  Conclusion  :  the  terms  of  the  Conclusion 

are  sometimes  called  extremes. 
Every  sy.,0.       2d.  Every  syllogism  has  three  and  only  three 


CHAP.   III.]  ANALYTICAL     OUTLINE.  89 


propositions;  viz.  the  major  premiss  ;  the  minor     gismhas 

.  ,  three  and 

premiss;  and  the  conclusion.  only  throe 

3d.    If  the  middle    term  is  ambiguous,    there  Pr°P°sitk)Ils- 

Middle  term 

are  in  reality  two  middle  terms,  in  sense,  though  must  not  be 
but  one  in  sound.  ambiguou9' 


There  are  two  cases  of  ambiguity:  1st.  Where 
the  middle  term  is  equivocal  ;  that  is,  when  used     l8tcase. 
in  different  senses   in  the  two  premises.      For 
example  : 


"  Light  is  contrary  to  darkness  ; 

J  Example. 

Feathers  are  light ;  therefore, 

Feathers  are  contrary  to  darkness." 


2d.  Where  the  middle  term  is  not  distrib 
uted  ;  for  as  it  is  then  used  to  stand  for  a  part 
only  of  its  significates,  it  may  happen  that  one 
of  the  extremes  is  compared  with  one  part  of 
the  whole  term,  and  the  other  with  another  part 
of  it.  For  example  : 

"  White  is  a  color ; 
Black  is  a  color  ;  therefore, 

Examples. 

Black  is  white." 
Again  : 

"  Some  animals  are  beasts  ; 
Some  animals  are  birds  ;  therefore, 
Some  birds  are  beasts." 

The  middle 

3d.    The  middle  term,  therefore,  must  be  dis-  term  must  be 

once  distrib- 

tributed,  once,  at  least,  in  the  premises  ;  that  is,       uted; 


LOGIC.  [BOOK  i. 


and  once  is  by  being  the  subject  of  a  universal,*  or  predi 
cate  of  a  negative  ;f  and  once  is  sufficient ; 
since  if  one  extreme  has  been  compared  with  a 
part  of  the  middle  term,  and  another  to  the 
whole  of  it,  they  must  have  been  compared  with 
the  same. 
Notermmust  4th.  No  term  must  be  distributed  in  the  con- 

be  distribu-  .  7.7 

ted  in  the  elusion  which  was  not  distributed  in  one  of  the 
wTiciTw™  Premises;  for,  that  would  be  to  employ  the 
not  distribu-  whole  of  a  term  in  the  conclusion,  when  you 

ted  in  a  y 

premiss,  had  employed  only  a  part  of  it  in  the  premiss  ; 
thus,  in  reality,  to  introduce  a  fourth  term. 
This  is  called  an  illicit  process  either  of  the 
major  or  minor  term.  J  For  example  : 

Example.  "  A^  quadrupeds  are  animals, 

A  bird  is  not  a  quadruped  ;  therefore, 

It  is  not  an  animal."     Illicit  process  of  the  major. 

Negative         5th.    From  negative  premises   you  can  infer 

premises 

prove  noth-  nothing.  For,  in  them  the  Middle  is  pronounced 
to  disagree  with  both  extremes;  therefore  they 
cannot  be  compared  together :  for,  the  extremes 
can  only  be  compared  when  the  middle  agrees 
with  both ;  or,  agrees  with  one,  and  disagrees 
with  the  other.  For  example  : 

Example.  «  A  fish  is  not  a  quadruped  ;" 

"  A  bird  is  not  a  quadruped,"  proves  nothing. 

*  Section  62.         f  Section  63.         \  Section  40. 


CHAP.   III.]  ANALYTICAL     OUTLINE.  91 


6th.    If  one  premiss  be  negative,   the  conclu-  ifoneprem- 

7  .  r         .          ,  ,          iss  is  nega- 

swn  must  be  negative;  lor,  in  that  premiss  the     ti     the 
middle  term  is  pronounced  to  disagree  with  one    c"™lus1011 

will  be  nega- 

of  the  extremes,  and  in  the  other  premiss  (which       tive; 
of  course  is  affirmative  by  the  preceding  rule), 
to  agree  with  the  other  extreme ;  therefore,  the 
extremes  disagreeing  with  each  other,  the  con 
clusion  is  negative.     In  the  same  manner  it  may   andretipro 
be  shown,  that  to  prove  a  negative  conclusion, 
one  of  the  premises  must  be  a  negative. 

By  these  six  rules   all   Syllogisms   are  to  be    what  fol 
lows  from 
tried;    and  from  them  it  will  be    evident,    1st,     these aix 

that  nothing  can  be  proved  from  two  particular 
premises;  (since  you  will  then  have  either  the 
middle  term  undistributed,  or  an  illicit  process. 
For  example : 

"  Some  animals  are  sagacious  ; 
Some  beasts  are  not  sagacious ; 
Some  beasts  are  not  animals.") 

And,  for  the  same  reason,  2dly,  that  if  one  of  2d  inference, 
the  premises  be  particular,  the  conclusion  must 
be  particular.     For  example  : 

"  All  who  fight  bravely  deserve  reward ;  -Exam  le 

"  Some  soldiers  fight  bravely  ;"  you  can  only  infer  that 
"  Some  soldiers  deserve  reward  :" 

for  to  infer  a  universal  conclusion  would  be 
an  illicit  process  of  the  minor.  But  from  two 


92 


LOGIC.  [BOOK  i. 


-  universal  Premises  you  cannot  always  infer  a 

*  universal  Conclusion.     For  example  : 

give  a  uni 
versal  con-  "  All  gold  is  precious  ; 
elusion.  All  gold  is  a  mineral ;  therefore, 
Some  mineral  is  precious.' 

And  even  when  we  can  infer  a  universal,  we 
are  always  at  liberty  to  infer  a  particular ;  since 
what  is  predicated  of  all  may  of  course  be  pre 
dicated  of  some. 


0  F     FAL  LACIES. 

Definition  of      §  68.    By  a  fallacy  is  commonly  understood 

a  fallacy. 

"  any  unsound  mode  of  arguing,  which  appears 

to  demand  our  conviction,  and  to  be  decisive 

of  the  question  in  hand,  when  in  fairness  it  is 

Detection  of,  not."     In  the  practical  detection  of  each  indi- 

acuteness.  vidual  fallacy,  much  must  depend  on  natural 
and  acquired  acuteness ;  nor  can  any  rules  be 
given,  the  mere  learning  of  which  will  enable 
us  to  apply  them  with  mechanical  certainty  and 

Hints  and  readiness ;  but  still  we  may  give  some  hints  that 
will  lead  to  correct  general  views  of  the  subject, 
and  tend  to  engender  such  a  habit  of  mind,  as 
will  lead  to  critical  examinations. 

same  of  LO-       Indeed,  the  case  is  the  same  with  respect  to 

gicingeneral.   T  ,  ,  i  i      • 

Logic  in  general;   scarcely  any  one  would,  in 
ordinary   practice,    state   to   himself   either   his 


CHAP.  III.]  ANALYTICAL     OUTLINE.  93 


own  or  another's  reasoning,  in  syllogisms  at  full   Logic  tends 

.       1  .    ,  to  cultivate 

length  ;  yet  a  familiarity  with  logical  principles     habits  of 

tends  very  much  (as  all  feel,  who  are  really  well  cle^90»- 

acquainted  with  them)  to  beget  a  habit  of  clear 

and  sound  reasoning.     The  truth  is,  in  this  as 

in  many  other  things,  there  are  processes  going    The  habit 

fixed,  we 

on  in  the  mind   (when  we    are  practising  any  naturally  foi- 

v  .   ,  .  , .  low  the 

thing  quite  familiar  to  us),  with   such  rapidity    processe8. 
as  to  leave  no  trace  in  the  memory ;  and  we 
often  apply  principles  which  did  not,  as  far  as 
we  are  conscious,  even  occur  to  us  at  the  time. 


§  69.   Let  it  be   remembered,   that   in   every   conclusion 

.  follows  from 

process  of  reasoning,  logically  stated,  the  con-   twoantece. 
elusion  is  inferred  from  two  antecedent  propo-   dentPrem- 

ises. 

sitions,  called  the  Premises.     Hence,  it  is  man 
ifest,  that  in  every  argument,  the  fault,  if  there    Fallacy,  if 

any,  either  in 

be  any,  must  be  either,  the  promises 

1st.  In  the  premises;  or, 

2d.  In  the  conclusion  (when  it  does  not  follow    or  conclu 
sion,  or  both. 
from  them) ;  or, 

3d.  In  both. 

In  every  fallacy,  the  conclusion  either  does  or 
does  not  follow  from  the  premises. 

When  the  fault  is  in  the  premises ;  that  is,  when  in  the 
when  they  are  such  as  ought  not  to  have  been 
assumed,  and  the  conclusion  legitimately  follows 
from  them,  the  fallacy  is  called  a  Material  Fal- 


94  LOGIC.  [HOOK  i. 

lacy,  because  it  lies  in  the  matter  of  the  argu 
ment. 

when  in  the      Where  the  conclusion  does  not  follow  from 
ion'   the  premises,  it  is  manifest  that  the  fault  is   in 
the  reasoning,  and  in  that  alone :  these,  there 
fore,  are  called  Logical  Fallacies,  as  being  prop 
erly  violations  of  those  rules  of  reasoning  which 
it  is  the  province  of  logic  to  lay  down, 
when  in         When  the  fault  lies  in  both  the  premises  arid 
reasoning,  the  fallacy  is  both  Material  and  Logical. 

Rules  for         §  70.  In  examining  a  train  of  argumentation, 

examining  a  -        •  c          r  11  i  •  . 

train  of  ar-    to  ascertain  if  a  fallacy  have  crept  into  it,  the 

gument.  following  points  would  naturally  suggest  them 
selves  : 

1st  Rule.  1st.  What  is  the  proposition  to  be  proved? 
On  what  facts  or  truths,  as  premises,  is  the  ar 
gument  to  rest  ?  and,  What  are  the  marks  of 
truth  by  which  the  conclusion  may  be  known  ? 

sdRuie.  2d.  Are  the  premises  both  true?  If  facts,  are 
they  substantiated  by  sufficient  proofs  ?  If  truths, 
were  they  logically  inferred,  and  from  correct 
premises  ? 

3d  Rule.  3d.  Is  the  middle  term  what  it  should  be,  and 
the  conclusion  logically  inferred  from  the  prem 
ises  ? 

suggestions       These  general  suggestions  may  serve  as  guides 

serve  as 

guides,      m  examining  arguments  for  the  purpose  of  de- 


CHAP.   III.]  ANALYTICAL     OUTLINE.  95 

tecting   fallacies ;    but  however  perfect  general     to  detect 
rules  may  be,  it  is  quite  certain  that  error,  in 
its  thousand  forms,  will  not  always  be  separated 
from  truth,  even  by  those  who  most  thoroughly 
understand  and  carefully  apply  such  rules. 


CONCLUDING     REMARKS. 

§  71.  The  imperfect  and  irregular  sketch  which      Logic 

corresponds 

has  here  been  attempted  of  deductive  logic,  may     with  the 


suffice  to  point  out  the  general  drift  and  purpose  r^ 
of  the  science,  and  to  show  its  entire  correspond 
ence  with  the  reasonings  in   Geometry.      The 
analytical  form,  which   has  here  been  adopted,    Analytical 
is,  generally  speaking,  better  suited  for  introdu 
cing  any  science  in  the  plainest  and  most  inter 
esting  form  ;  though  the  synthetical  is  the  more   synthetical 
regular,  and  the  more  compendious  form  for  sto 
ring  it  up  in  the  memory. 


§  72.  It  has  been  a  matter  about  which  wri-    induction: 

does  it  form 

ters  on  logic  have  differed,  whether,  and  in  con-     a  part  Of 
formity  to   what   principles,  Induction   forms   a 


part  of  the  science  ;  Archbishop  Whately  main- 
taining  that  logic  is  only  concerned  in  inferring 
truths  from  known  and  admitted  premises,  and 
that  all  reasoning,  whether  Inductive  or  Deduc 
tive,  is  shown  by  analysis  to  have  the  syllogism 


96  LOGIC.  [BOOK  i. 

Mill's  views,  for  its  type  ;  while  Mr.  Mill,  a  writer  of  perhaps 
greater  authority,  holds  that  deductive  logic  is 
but  the  carrying  out  of  what  induction  begins ; 
that  all  reasoning  is  founded  on  principles  of  in 
ference  ulterior  to  the  syllogism,  and  that  the 
syllogism  is  the  test  of  deduction  only. 

Without   presuming   at  all    to   decide  defini 
tively  a  question  which  has  been  considered  and 

Reasons  for  passed  upon  by  two  of  the  most  acute  minds  of 

the  course 

taken.  the  age,  it  may  perhaps  not  be  out  of  place  to 
state  the  reasons  which  induced  me  to  adopt 
the  opinions  of  Mr.  Mill  in  view  of  the  par 
ticular  use  which  I  wished  to  make  of  logic. 

Leading  ob-       §  73.    It  was,  as    stated  in  the  general  plan, 

jects  of  the 

plan-      one  of  my  leading  objects  to  point  out  the  cor 
respondence  between  the  science  of  logic  and 
the  science  of  mathematics  :   to  show,   in  fact, 
Toshowthat  that  mathematical  reasoning  conforms,  in  every 

mathemati-  .  ,,  . 

cai  reasoning  respect,  to  the  strictest  rules  oi  logic,  and  is  m- 
conformsto  deed  ^ut  ]Og[c  applied  to  the  abstract  quantities, 

logical  rules. 

Number  and  Space.     In  treating  of  space,  about 

which  the  science  of  Geometry  is  conversant,  we 

shall  see  that  the  reasoning  rests  mainly  on  the 

Axioms,  how  axioms,  and  that  these  are  established  by  induc- 

established.     ..  -T,, 

tive  processes.  The  processes  of  reasoning  which 
relate  to  numbers,  whether  the  numbers  are  rep 
resented  by  figures  or  letters,  consist  of  two  parts. 


CHAP.  III.]  ANALYTICAL     OUTLINE.  97 

1st.  To  obtain  formulas  for,  that  is,  to  express 
in  the  language  of  science,  the  relations  between 
the  quantities,  facts,  truths  or  principles,  what-  Two  Parts  °f 

the  reasoning 

ever  they  may  be,  that  form  the  subject  of  the     process, 
reasoning ;  and, 

2dly.  To  deduce  from  these,  by  processes 
purely  logical,  all  the  truths  which  are  implied 
in  them,  as  premises. 


§  74.    Before  dismissing  the   subject,   it  may    Aiiinduc- 
be  well  to  remark,   that  every  induction   may  thrown  into 
be  thrown  into  the  form  of  a  syllogism,  by  sup-     .^Jj* 
plying  the  major  premiss.     If  this  be  done,  we  syiiogism,by 

admitting  a 

shall  see  that  something  equivalent  to  the  uni-  proper  major 

formity  of  the  course  of  nature  will  appear  as 

the  ultimate   major  premiss   of  all    inductions  ; 

and  will,   therefore,   stand  to  all  inductions   in 

the  relation  in  which,  as  has  been  shown,  the 

major  premiss  of  a  syllogism  always  stands   to 

the  conclusion ;  not  contributing  at  all  to  prove 

it,  but  being  a  necessary  condition  of  its  being 

proved.     This  fact  sustains  the  view  taken  by 

Mr.  Mill,  as  stated  above ;  for,  this  ultimate  ma-    HOW  this 

:   i     •  i       •        •         r       •       •  •    r         major  prem- 

jor  premiss,  or  any  substitution  tor  it,  is  an  inter-  issisobtain. 
ence  by  Induction,  but  cannot  be  arrived  at  by       ed- 
means  of  a  syllogism. 

7 

I 


BOOK    II. 

MATHEMATICAL    SCIENCE, 


CHAPTER    I. 

QUANTITY   AND    MATHEMATICAL     SCIENCE     DEFINED DIFFERENT   KINDS    OF    QUAN 
TITY LANGUAGE    OF   MATHEMATICS    EXPLAINED SUBJECTS    CLASSIFIED UNIT 

OF   MEASURE    DEFINED — MATHEMATICS   A    DEDUCTIVE    SCIENCE. 


QUANTITY. 


§  75.  QUANTITY  is  a  general  term  applicable     Quantity 


defined. 


to  every  thing  which  can  be  increased  or  dimin 
ished,  and  measured.  There  are  two  kinds  of 
quantity : 

1st.  Abstract  Quantity,  or  quantity,  the  con-    Abstract. 
ception  of  which  does  not  involve  the  idea  of 
matter;  and, 

2dly.    Concrete    Quantity,    which    embraces    concrete. 
every  thing  that  is  material. 

§  76.  Mathematics  is  the  science  of  quantity ;  Mathematics 
that  is,  the  science  which  treats  of  the  measures 
of  quantities  and  their  relations  to  each  other. 
It  is  divided  into  two  parts : 


100 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


Pure  1st.    The  Pure  Mathematics,  embracing  the 

Mathematics.  princip}es  of  the  science,  and  all   explanations 

of  the  processes  by  which  those  principles  are 

derived  from  the  laws  of  the  abstract  quantities, 

Number  and  Space  ;  and, 

Mixed          2d.  The  Mixed  Mathematics,  embracing  the 

Mathematics.  appijcations  of  those  principles  to  all  investiga 

tions  and  to  the  solution  of  all  questions  of  a 

practical  nature,  whether  they  relate  to  abstract 

or  concrete  quantity. 

Mathematics,      §  77.    Mathematics,  in   its  primary  significa- 

tto  ancients:  ti°n>  as  usea<   by  the  ancients,  embraced  every 

acquired  science,  and  was  equally  applicable  to 

all  branches  of  knowledge.     Subsequently  it  was 

restricted  to  those   branches   only  which  were 

acquired  by  severe  study,  or  discipline,  and  its 

embraced  aii  votaries  were  called  Disciples.     Those'  subjects, 

subjects         IP  1-1  • 

which  were  therefore,  which  required  patient  investigation, 


exact  reasoning,  and  the  aid  of  the  mathemati- 
ture.  ca]  analysis,  were  called  Disciplinal  or  Mathe 
matical,  because  of  the  greater  evidence  in  the 
arguments,  the  infallible  certainty  of  the  conclu 
sions,  and  the  mental  training  and  development 
which  such  exercises  produced. 

Pure  §  78.   It  has  already  been  observed  that    the 

-i  pure  Mathematics  embrace  all  the  principles  of 


CHAP.  I.]  NUMBER.  101 


the  science,   and  that  these   principles   are  de-    what  they 
duced,  by  processes  of  reasoning  upon  the  two 
abstract   quantities,   Number   and    Space.      All 
the  definitions   and   axioms,   and  all  the  truths 
deduced  from  them,  are  traceable  to  those  two 

sources.     Here,  then,  two   important   questions    TWO  ques 
tions, 
present  themselves : 

1st.  How  are  we  to  attain  a  clear  and  true  Howdowe 

conceive  of 

conception  of  these  quantities  ?   and,  the  quanti 

ties? 
2dly.  How  are  we  to  represent  them,  and  what  How  re  re_ 

language  are  we  to  employ,  so  as  to  make  their   ^t116111- 
properties  and  relations   subjects   of  investiga 
tion  ? 

NUMBER. 

§  79.  NUMBERS  are  expressions  for  one  or  Number 
more  things  of  the  same  kind.  How  do  we  defined, 
attain  unto  the  significations  of  such  expres- 


sions  ?     By  first  presenting  to  the  mind,  through  tein  m  idea 

*  of  number. 

the    eye,    a    single   thing,    and   calling    it    ONE. 
Then  presenting  two  things,  and  naming  them 
TWO  :  then  three  things,  and  naming  them  THREE  ; 
and  so  on  for  other  numbers.     Thus,  we  acquire 
primarily,   in  a   concrete   form,    our   elementary  it  ja  done  by 
notions  of  number,   by  perception,  comparison,  £m%lon, 
and  reflection ;  for,  we  must  first  perceive  how       ^ 

reflection. 

many  things  are  numbered ;  then  compare  what 

is  designated  by  the   word   one,  with   what   is     Reasons. 


102  MATHEMATICAL     SCIENCE.  [BOOK  II. 

designated  by  the  words  two,   three,    &c.,  and 

then  reflect  on  the  results  of  such  comparisons 

until  we  clearly  apprehend  the  difference  in  the 

signification  of  the  words.    Having  thus  acquired, 

in  a  concrete  form,  our  conceptions  of  numbers, 

we  can  consider  numbers  as  separated  from  any 

particular  objects,  and  thus  form  a  conception 

TWO  axioms  of  them  in  the  abstract.     We  require  but  two 

the  formation  axioms  for  the  formation  of  all  numbers  : 

ofnumbers. 


1st  axiom,  and  that  the  number  which  results  will  be  great 
er  by  one  than  the  number  to  which  the  one 
was  added. 

2d  axiom.  2d.  That  one  may  be  divided  into  any  num 
ber  of  equal  parts. 

Language        §  80.    But  what  language  are  we  to  employ 

employed. 

as  best  suited  to  furnish  instruments  of  thought, 
and  the  means  of  recording  our  ideas  and  ex- 
Theten  pressing  them  to  others?  The  ten  characters, 
called  figures,  are  the  alphabet  of  this  language, 
and  the  various  ways  in  which  they  are  com 
bined  will  be  fully  explained  under  the  head 
Arithmetic,  a  chapter  devoted  to  the  considera 
tion  of  numbers,  their  laws  and  language. 


CHAP.  I.]  SPACE.  103 


SPACE. 

§  81.  SPACE  is  indefinite  extension.     We  ac-      sPace 

defined. 

quire  our  ideas  of  it  by  observing  that  parts  of 
it  are  occupied  by  matter  or  bodies.     This  ena 
bles  us  to  attach  a  definite  idea  to  the  word 
place.      We  are  then  able  to   say,  intelligibly,      Place: 
that  a  point  is  that  which  has  place,  or  position     a  point- 
in  space,  without  occupying  any  part  of  it.    Hav 
ing  conceived  a  second  point  in  space,  we  can 
understand    the   important  axiom,   "  A   straight 
line  is  the  shortest  distance  between  two  points ;"  Axiom  con- 
and  this  line  we  call  length  or  a  dimension  of  J^^  ime. 
space, 

§  82.  If  we  conceive  a  second  straight  line 
to  be  drawn,  meeting  the  first,  but  lying  in  a 
direction  directly  from  it,  we  shall  have  a  second 
dimension  of  space,  which  we  call  breadth.  If  Breadth 

defined. 

these  lines  be  prolonged,  in  both  directions,  they 
will  include  four  portions  of  space,  which  make 
up   what   is    called  a   plane   surface   or  plane : 
hence,  a  plane  has  two  dimensions,  length  and     A  plane 
breadth.     If  now  we  draw  a  line  on  either  side 
of  this  plane,  we  shall  have  another  dimension  of 
space,  called  thickness:  hence,  space  has  three    space  has 
dimensions — length,  breadth,  and  thickness.  siongt 


104  MATHEMATICAL     SCIENCE.  [flOOK  II. 

Figure          §  83.  A  portion  of  space  limited  by  bounda- 
deflned.     ^^  .g  cajje(j  a  pigure.     If  such  portion  of  space 

Line  defined  have  but  one  dimension,  it  is  called  a  line,  and 
may  be  limited  by  two  points,  one  at  each  ex- 
Two  kinds  of  tremity.     There  are  two  kinds  of  lines,  straight 
•MUM  and  anc^  curve<^-     ^  straignt  lme>  *s  one  which  does 
curved.     not  change  ^s  direction  between  any  two  of  its 
points,  and  a  curved  line  constantly  changes  its 
direction  at  every  point. 


§  84.  A  portion  of  space  having  two  dimen- 

faurface :  1 

sions  is  called  a  surface.     There  are  two  kinds 

Plane, 

curved,     of  surfaces — Plane  Surfaces   and   Curved   Sur 
faces.     With  the  former,  a  straight  line,  having 

Difference. 

two   points   in   common,   will    always    coincide, 
however  it  may  be  placed,  while  with  the  latter 

Boundaries 

of  a  surface,  it   will   not.      The   boundaries   of  surfaces   are 
lines,  straight  or  curved. 


%iid denned  $  85'  ^  Por^on  °^  sPace  having  three  dimen 
sions,  is  called  a  solid,  and  solids  are  bounded 
either  by  plane  or  curved  surfaces. 


§  86.  The  definitions  and  axioms  relating  to 
space,  and  all  the  reasonings  founded  on  them, 
science  of    make  up  the  science  of  Geometry.     They  will 
all  be  fully  treated  under  that  head. 


CHAP.   I.]  ANALYSIS.  105 


ANALYSIS. 

§  87.  ANALYSIS  is  a  general  term  embracing     Analysis. 
all  the  operations  which  can  be  performed  on 
quantities  when  represented  by  letters.     In  this 
branch  of  mathematics,   all  the  quantities  con 
sidered,  whether  abstract  or  concrete,  are  rep-    Quantities 
resented   by   letters    of    the    alphabet,    and   the    b  letJJ 
operations  to  be  performed  on  them  are  indi 
cated   by  a   few  arbitrary  signs.      The   letters 
and  signs  are  called  Symbols,  and  by  their  com-     symbols, 
bi nation  we  form  the  Algebraic   Notation   and 
Language. 


§  88.  Analysis,  in  its  simplest  form,  takes  the     Analysis> 

Algebra; 

name  of  Algebra ;  Analytical  Geometry,  the  Dif-    Analytical 
ferential  and  Integral  Calculus,  extended  to  in 
clude  the  Theory  of  Variations,  are   its  higher 
and  most  advanced  branches. 


§  89.  The  term  Analysis  has  also  another  sig-  Term 

.  .sis  defined. 

nmcation.     It  denotes  the  process  of  separating 
any  complex  whole  into  the  elements  of  which    its  nature, 
it  is  composed.      It  is  opposed  to  Synthesis,  a     synthesis 
term  which  denotes  the  processes  of  first  con 
sidering  the  elements  separately,  then  combining 
them,  and  ascertaining  the  results  of  the  combi 
nation. 


106  MATHEMATICAL     SCIENCE.  [BOOK  II. 


Analytical        The  Analytical  method  is  best  adapted  to  in 
vestigation,  and  the  presentation  of  subjects  in 
synthetical    their  general  outlines;    the  Synthetical  method 

method.  .  1 

is  best  adapted  to  instruction,  because  A  exhib 
its  all  the  parts  of  a  subject  separately,  and  in 
their  proper  order  and  connection.  Analysis 
deduces  all  the  parts  from  a  whole :  Synthesis 
forms  a  whole  from  the  separate  parts. 


Arithmetic,       §  go.  Arithmetic,  Algebra,  and  Geometry  are 

Algebra, 

Geometry,  the  elementary  branches  of  Mathematical  Sci- 
ence-  Every  truth  which  is  established  by 
mathematical  reasoning,  is  developed  by  an 
arithmetical,  geometrical,  or  analytical  process, 
or  by  a  combination  of  them.  The  reasoning 
in  each  branch  is  conducted  on  principles  iden 
tically  the  same.  Every  sign,  or  symbol,  or 
technical  word,  is  accurately  defined,  so  that  to 
each  there  is  attached  a  definite  and  precise 
idea.  Thus,  the  language  is  made  so  exact  and 
certain,  as  to  admit  of  no  ambiguity. 


LANGUAGE     OF    MATHEMATICS. 

§  91.  The  language  of  Mathematics  is  mixed, 
mixed.      Although  composed   mainly  of  symbols,   which 
are  defined  with  reference  to   the    uses   which 
are  made  of  them,   and  therefore  have   a  pre- 


CHAP.   I.]          LANGUAGE     OF     MATHEMATICS.  107 

else  signification  ;  it  is  also  composed,  in  part, 
of  words  transferred  from  our  common  language. 
The  symbols,  although  arbitrary  signs,  are,  nev-  symbols 

general. 

ertheless,  entirely  general,  as  signs  and  instru 
ments  of  thought  ;  and  when  the  sense  in  which 
they  are  used  is  once  fixed,  by  definition,  they 
preserve  throughout  the  entire  analysis  precise 
ly  the  same  signification.  The  meaning  of  the  words  bor- 

.    -  .       rowed  from 

words  borrowed  trom  our  common  vocabulary  is     Comm0n 


often  modified,  and  sometimes  entirely  changed, 

J  are  modilied 

when  the  words  are  transferred  to  the  language  and  used  in  a 

technical 

of  science.     They  are  then  used  in  a  particular      sense. 
sense,  and  are  said  to  have  a  technical  significa 
tion. 


§  92.    It  is  of  the  first  importance  that  the    Language 
elements  of  the  language  be  clearly  understood, 


—  that  the  signification  of  every  wrord  or  sym 

bol  be  distinctly  apprehended,  and  that  the  con-  , 

nection  between  the  thought  and  the  word  or 

symbol  which  expresses  it  be  so  well  established 

that  the  one  shall  immediately  suggest  the  other. 

It  is  not  possible  to  pursue  the  subtle  reasonings    Mathemati- 

f    cal  reason- 

of  Mathematics,  and  to  carry  out  the  trams  ot  ^require 
thought  to  which  they  give  rise,  without  entire 
familiarity  with   those   means  which   the    mind 
employs  to   aid  its   investigations.      The   child   canr.otuae 

any  language 

cannot  read  till  he  has   learned   the    alphabet; 


108  MATHEMATICAL     SCIENCE.  [BOOK  II. 

well  till  we   nor  can  the  scholar  feel  the  delicate  beauties  of 

know  it.     ghakspeare,  or  be  moved   by  the   sublimity  of 

Milton,  before   studying  and   learning  the    lan 

guage   in   which   their   immortal    thoughts    are 

clothed. 

Quantities        §  93.  All  Quantities,  whether  abstract  or  con- 

are  repre 

sented  by    crete,   are,   in    mathematical    science,   presented 

•iKUreoper-  to  tne  mmd   by  arbitrary  symbols.      They    are 

these?  m-    v^ewec^  anc^  operated  on  through  these  symbols 

bois.       which  represent   them;    and   all  operations  are 

indicated    by   another   class    of  "symbols    called 

signs.       signs.       These,    combined    with    the    symbols 

Whatconsti  wmc^    represent    the    quantities,    make    up,    as 

tutesthe    we    have  stated  above,  the   pure   mathematical 

language. 

language  ;     and   this,    in    connection    with    that 

which  is  Borrowed  from  our  common  language, 

forms    the    language    of  mathematical    science. 

^  This  language    is    at   once    comprehensive    and 

its  nature,    accurate.      It   is  capable   of    stating   the   most 

general  proposition,  and  presenting  to  the  mind, 

in  their  proper  order,  every  elementary  princi- 

whatitac-  ple  connected  with  its  solution.     By  its  gener- 

omphshes. 


pure  and  mixed  sciences,  and  gathers  into  con 
densed  forms  all  the  conditions  and  relations 
necessary  to  the  development  of  particular  facts 
and  universal  truths  ;  and  thus,  the  skill  of  the 


CHAP.   I.]  QUANTITY     MEASURED.  109 

analyst  deduces  from  the  same  equation  the  ve-   Extent  and 
locity  of  an  apple  falling  to  the  ground,  and  the     Anli^is. 
verification  of  the  law  of  universal  gravitation. 

QUANTITY     MEASURED. 

§  94.  Quantity  has  been  defined,  "  any  thing    Quantity, 
which  can  be  increased  or  diminished,  and  meas 
ured."     The  terms  increased  or  diminished,  are    increased 

and 

easily  understood,  implying  merely  the  property  diminished, 
of  being   made  larger   or   smaller.      The   term 
measured  is  not  so  easily  explained,  because  it 
has  only  a  "relative  meaning. 

The  term  "  measured,"  applied  to  a  quantity,    Measured, 
implies  the  existence  of  some   known   quantity 
of  the  same  kind,  which  is  regarded  as  a  stand-     what  it 
ard,  and  with  which  the  quantity  to  be  meas 
ured  is   compared  with  respect  to  its  extent  or 
magnitude.     To  such  standard,  whatever  it  may    standard: 
be,  we  give  the  name  of  unity,  or  unit  of  meas-     is  called 

unity. 

ure ;  and  the  number  of  times  which  any  quan 
tity  contains  its  unit  of  measure,  is  the  numerical 
value  of  the  quantity  measured.  The  extent 
or  magnitude  of  a  quantity  is,  therefore,  merely  Magnitude: 

,,  •  i  r   •        merely  rela- 

relative,  and  hence,  we  can  form  no  idea  of  it,       tive. 
except  by  the  aid  of  comparison.      Space,  for 
example,  is  entirely  indefinite,  and  we  measure      space: 
parts  of  it  by  means  of  certain  standards,  called 


110  MATHEMATICAL     SCIENCE.  [BOOK  II. 


Measurement  measures  ;  and  after  any  measurement  is  com- 

ascertains  re-      ,  . 

lation:      pletcd,  we  have  only  ascertained  the  relation  or 

proportion  which  exists  between  the  standard  we 

n  processor  adopted  and  the  thing  measured.  Hence,  measure- 

cumparison. 

ment  is,  after  all,  but  a  mere  process  of  comparison. 
weight  and       §  95.    The    abstract    quantities,    Weight  and 

velocity  :       Tr    , 

known  by  Velocity,  are  but  vague  and  indefinite  concep- 
fa^  untQ  compared  with  their  units  of  meas 
ure,  and  even  these  are  arrived  at  only  by  pro- 
cesses  of  comparison.  Indeed,  most  of  our 

a  general 

.ne.uod.  knowledge  of  all  subjects  is  obtained  in  the 
same  way.  We  compare  together,  very  care 
fully,  all  the  facts  which  form  the  basis  of  an 
induction;  and  we  rely  on  the  comparison  of 
the  terms  in  the  major  and  minor  premises  for 
every  conclusion  by  a  deductive  process. 

Quantity.  §  96.  Quantity,  as  we  have  seen,  is  divided 
into  Abstract  and  Concrete—  the  abstract  quan- 

Abstract.  tity  being  a  mere  mental  conception,  having 
for  its  sign  a  number,  a  letter,  or  a  geometrical 

concrete,  figure.  A  concrete  quantity  is  a  physical  ob 
ject,  or  a  collection  of  such  objects,  and  may 
likewise  be  represented  by  numbers,  letters,  or 

,          , 

by  the  geometrical  magnitudes  regarded  as  ma- 
terial  The  number  "  th^ee"  is  entirely  abstract, 
expressing  an  idea  having  no  connection  with 


sented. 


CHAP.   I.]  PURE     MATHEMATICS.  Ill 

material  things  ;  while  the  number  "  three  pounds 
of  tea,"  or  "  three  apples/'  presents  to  the  mind 
an  idea  of  physical  objects.  So,  a  portion  of 
space,  bounded  by  a  surface,  all  the  points  of 
which  are  equally  distant  from  a  certain  point 
within  called  the  centre,  is  but  a  mental  con 
ception  of  form;  but  regarded  as  a  solid  mass,  ofthecon- 

...  . 

it  gives  rise  to  the  additional  idea  of  a  material 
substance. 


PURE     MATHEMATICS. 

§  97.    The   Pure  Mathematics  are   based  on       Pure 

Mathematics: 

definitions    and  intuitive  truths,    called   axioms, 

which  are  inferred  from  observation  and  expe-  what  are  its 

foundations. 

rience ;  that  is,  observation  and  experience  fur 
nish  the  information  necessary  to  such  intuitive 
inductions.*  From  these  definitions  and  axioms, 
as  premises,  all  the  truths  of  the  science  are  estab 
lished  by  processes  of  deductive  reasoning ;  and 
there  is  not,  in  the  whole  range  of  mathemat-  Itste8tsof 

*  truths: 

ical  science  any  logical  test  of  truth,   but  in  a 
conformity  of  the  conclusions  to  the  definitions   w  ^  e' 
and  axioms,  or  to  such  principles  as  have  been 
established   from   them.      Hence,  we    see,    that  in  what  the 

science  con- 

the   science   of  Pure  Mathematics,  which  con-       gists. 
sists  merely  in  inferring,  by  fixed  rules,  all  the 


*  Section  27. 


112  MATHEMATICAL     SCIENCE.  [BOOK   II. 


is  purely     truths  which  can  be  deduced  from  given  prem- 
Deductive-    ises,  is  purely  a  Deductive  Science.     The  pre 
cision  and  accuracy  of  the  definitions  ;  the  cer 
tainty  which  is  felt  in  the  truth  of  the  axioms  ; 
Precision  of  the  obvious  and  fixed  relation  between  the  sign 
e'  and  the  thing  signified ;    and  the   certain    for 
mulas  to  which  the  reasoning  processes  are  re 
duced,  have  given  to  mathematics  the  name  of 


Exact  .  ,? 

Science.       "  Exact  Science. 


AII  reasoning      §  98.  We  have  remarked  that  all  the  reason- 

^lUion^"  ings  of  mathematical  science,  and  all  the  truths 

axioms.     which   they  establish,   are  based  on   the  defini 

tions  and  axioms,  which  correspond  to  the  major 

premiss  of  the  syllogism.      If  the  resemblance 

which  the  minor  premiss  asserts  to  the  middle 

Relations  not  term  were  obvious  to  the  senses,  as  it  is  in  the 

obvious.  .   .  ,, 

proposition,  "  feocrates  was  a  man,  or  were 
at  once  ascertainable  by  direct  observation,  or 
were  as  evident  as  the  intuitive  truth,  "  A  whole 
is  equal  to  the  sum  of  all  its  parts  ;"  there 
Deductive  would  be  no  necessity  for  trains  of  reasoning, 

Science, 

and  Deductive  Science  would  not  exist.     Trains 


Trains  of    of  reasoning  are  necessary  only  for  the  sake  of 

extending  the  definitions  and    axioms   to  other 

what  they    cases  in  which  we  not  only  cannot  directly  ob- 

accomplish. 

serve  what  is  to  be  proved,  but  cannot  directly 
observe  even  the  mark  which  is  to  prove  it. 


CHAP.   I.]  PURE     MATHEMATICS.  113 


§  99.  Although   the  syllogism  is  the  ultimate    syllogism, 
test  in  all  deductive  reasoning   (and  indeed   in  ^deduction. 
all  inductive,  if  we  admit  the  uniformity  of  the 
course  of  nature),  still  we  do  not  find  it  con 
venient  or  necessary,  in  mathematics,  to  throw 
every  proposition  into  the  form  of  a  syllogism. 

The  definitions   and   axioms,   and  the  propo-  Axioms  and 

i  v    i       i     /•  -    definitions, 

sitions  established  irom  them,   are  our  tests  oi  tests  or  truth: 
truth ;  and  whenever  any  new  proposition  can 

be   brought   to   conform    to   any   one   of   these    A  proposi 
tion  :  when 
tests,  it  is  regarded  as  proved,  and  declared  to     proved. 

be  true. 


§  100.    When   general    formulas  have   been     When  a 

principle 

framed,  determining  the  limits  within  which  the   maybe  re- 


deductions  may  be  drawn  (that   is,   what  shall 
be   the   tests   of  truth),  as  often  as  a  new  case 
can  be  at  once  seen  to  come  within  one  of  the 
formulas,  the  principle  applies  to  the  new  case, 
and  the  business  is  ended.     But  new  cases  are     Trains  of 
continually  arising,  which  do  not  obviously  come  ^"^ 
within  any  formula  that  will  settle  the  questions       sary- 
we  want  solved  in   regard   to   them,   and  it    is 
necessary    to   reduce    them    to    such    formulas. 
This  gives  rise  to  the  existence  of  the  science    They  give 
of  mathematics,  requiring  the  highest  scientific    "^^ 
genius  in  those  who  contributed  to  its  creation,  mathematics. 
and  calling  for  a  most  continued  and  vigorous 

8 


Hi  MATHEMATICAL     SCIENCE.  [BOOK  II. 

exertion  of  intellect,  in  order  to  appropriate  it. 
when  created. 

COMPARISON     OF     QUANTITIES. 

Mathematics       j  JQJ     \ye  nave  seen  that  the  pure  mathe- 

concerned 

with  Number  matics  are  concerned  with  the  two  abstract 
and  8pa«*.  qUantitjeSj  ]\Tumber  and  Space.  We  have  also 
Reasoning  seen  that  reasoning  necessarily  involves  coin- 
parison :  hence,  mathematical  reasoning  must 
consist  in  comparing  the  quantities  which  come 
from  Number  and  Space  with  each  other. 


§  102.    Any   two  quantities,    compared   with 

ties  can  sus 
tain  but  two  each  other,  must  necessarily  sustain  one  of  two 

relations  :  they  must  be  equal  or  unequal.  What 
axioms  or  formulas  have  we  for  inferring  the 
one  or  the  other  ? 


AXIOMS  OR  FORMULAS  FOR  INFERRING  EQUALITY. 

1.  Things  which  being  applied  to  each  other 
coincide,  are  equal  to  one  another. 

Formulas        2    Things  which  are  equal  to  the  same  thing 
Equality,     are  equal  to  one  another. 

3.  A  whole  is  equal  to  the  sum  of  all  its  parts. 

4.  If  equals  be  added  to  equals,  the  sums  are 
equal. 


CHAP.  I.]        COMPARISON    OF     aUANTITIES.  115 

5.  If  equals  be  taken  from  equals,  the  remain 
ders  are  equal. 

AXIOMS  OR  FORMULAS  FOR  INFERRING  INEQUALITY. 

1.  A  whole  is  greater  than  any  of  its  parts. 

2.  If  equals  be  added  to  unequals,  the  sums    Formulas 
are  unequal. 

Inequality. 

3.  If  equals  be  taken  from  unequals,  the  re 
mainders  are  unequal. 

§  103.  We  have  thus  completed  a  very  brief    outline  of 

i  i  1-1  r-     •**-      i  i   Mathematics 

and   general   analytical   view   of   Mathematical   completed. 
Science.      We   have  endeavored   to   point   out 
the  character  of  the  definitions,  and  the  sources 
as  well  as  the  nature  of  the  elementary  and  in 
tuitive  propositions  on  which  the  science  rests ;    What  fea 
tures  have 
the  kind  of  reasoning  employed  in  its  creation,       been 

and  its  divisions  resulting  from  the  use  of  dif 
ferent  symbols  and  differences  of  language.  We 
shall  now  proceed  to  treat  the  subjects  separ 
ately. 


rilAP.   II.]       ARITHMETIC FIRST     NOTIONS.  117 


CHAPTER    II. 


ARITHMETIC SCIENCE    AND    ART    OF    NUMBERS. 


SECTION   I. 


INTEGER    UNITS. 


FIRST    NOTIONS     OF    NUMBERS. 

§  104.  THERE  is  but  a  single  elementary  idea  But  one  eie- 

-  ,,  ...  mentary  idea 

in  the  science  of  numbers:  it  is  the  idea  of  the  in  numbers. 
UNIT  ONE.     There  is  but  one  way  of  impressing     HOW  im- 
this  idea  on  the  mind.     It  is  by  presenting  to 
the  senses  a  single  object ;    as,   one  apple,  one 
peach,  one  pear,  &c. 


§   105.    There   are  three  signs  by  means   of  Three  signs 

which  the  idea  of  one  is  expressed  and  commu-  ing  it. 
nicated.     They  are, 

1st.  The  word  ONE.  Aword- 

2d.  The  Roman  character  I.  Roman 

character: 

3d.  The  figure  1.  Flgure_ 


118  MATHEMATICAL     SCIENCE.  [BOOK  II. 


New  ideas        §  106.  If  one  be  added  to  one,  the  idea  thus 

by  adding    arising  is  different  from  the  idea  of  one,  and  is 

one'       complex.     This  new  idea  has  also  three  signs; 

viz.   TWO,  II.,   and  2.     If  one  be  again   added, 

that  is,  added  to  two,  the  new  idea  has  likewise 

three  signs ;  viz.  THREE,  III.,  and  3.      The  ex- 

Theexpres-  pressions  for  these,  and  similar  ideas,  are  called 

sions  are 

numbers,     numbers  i  hence, 

Numbers         NUMBERS    are    expressions  for   one   or   more 


defined. 


things  of  the  same  kind. 


IDEAS  OF  NUMBERS  GENERALIZED. 


ideas  of         §  107.  If  we  begin  with  the  idea  of  the  nurn- 

numbers      , 

generalized,  tor  one,  and  then  add  it  to  one,  making  two ; 
and  then  add  it  to  two,  making  three  ;  and  then 
to  three,  making  four ;  and  then  to  four,  making 
HOW  formed,  five,  and  so  on ;  it  is  plain  that  we  shall  form  a 
series  of  numbers,  each  of  which  will  be  greater 
unity  the    by  one  than  that  which  precedes  it.     Now,  one 
nwe     8  °r  unityj  is  t*le  basis  of  this  series  of  numbers, 
of  expressing  and  each   number   may  be   expressed  in   three 


them. 

ways  : 


1st  way.          1st.  By  the  words  ONE,  TWO,  THREE,  &c.,  of  our 

common  language  ; 

™ ™y-         2d.  By  the  Roman  characters;  and, 
3d  way.          3d.  By  figure s. 


CHAP.   II.]  ARITHMETIC UNITY.  119 


§  108.  Since  all  numbers,  whether  integer  or  AH  numbers 
fractional,  must  come  from,  and  hence  be  con-       one: 
nected  with,  the  unit  one,  it  follows  that  there 
is  but  one  purely  elementary  idea  in  the  science 
of  numbers.     Hence,  the  idea  of  every  number,    Hence  but 

.  one  idea  that 

regarded  as  made  up  01  units    (and  all  numbers  i9pureiyeie- 
except  one  must  be  so  regarded  when  we  ana 
lyze  them),  is  necessarily  complex.     For,  since     Another 
the  number  arises  from  the  addition  of  ones,  the    notions  are 

complex. 

apprehension  of  it  is  incomplete  until  we  under 
stand  how  those  additions  were  made  ;  and  there 
fore,  a  full  idea  of  the  number  is  necessarily 
complex. 


§  109.  But  if  we  regard  a  number  as  an  en 
tirety,  that  is,  as  an  entire  or  whole  thing,  as  an 
entire  two,  or  three,  or  four,  without  pausing  to  when  a 

-,....  .  number  may 

analyze  the  units  of  which  it  is  made  up,  it  may  be  regarded 
th'en  be  regarded  as  a  simple  or  incomplex  idea;  as" 
though,  as  we  have  seen,  such  idea  may  always 
be  traced  to  that  of  the  unit  one,  which  forms 
the  basis  of  the  number. 


UNITY     AND     A    UNIT     DEFINED. 

§  110.  When  we  name  a  number,  as  twenty  what  is  ne- 
feet,  two  things  are  necessary  to  its  clear  appre-  "^j^j 

of  a  number 


120  MATHEMATICAL     SCIENCE.  [BOOK  II. 

First.  1st.    A   distinct   apprehension    of    the    single 

thing  which  forms  the  basis  of  the  number  ;  and, 

second.         2d.  A  distinct  apprehension  of  the  number  of 

times  which  that  thing  is  taken. 
The  basis  of       The  single  thing,  which  forms  the  basis  of  the 

the  uumber  i  •  n     i  TA    •  n      i 

number,  is  called  UNITY,  or  a  UNIT.     It  is  called 


IS  UNITY. 


when  H  is   unity,  when  it  is  regarded  as  the  primary  basis 

called  UNITY,  Qf  the  number  ;  that  is,  when  it  is  the  final  stand 

ard  to  which  all  the  numbers  that  come  from  it 

and  when  a  are  referred.  It  is  called  a  unit  when  it  is  re 
garded  as  one  of  the  collection  of  several  equal 
things  which  form  a  number.  Thus,  in  the  ex 
ample,  one  foot,  regarded  as  a  standard  and  the 
basis  of  the  number,  is  called  UNITY  ;  but,  con 
sidered  as  one  of  the  twenty  equal  feet  which 
make  up  the  number,  it  is  called  a  UNIT. 

OF  SIMPLE  AND  DENOMINATE  NUMBERS.  « 

Abstract  §  111.  A  simple  or  abstract  unit,  is  ONE,  with 
out  regard  to  the  kind  of  thing  to  which  the  term 
one  may  be  applied. 

Denominate  A  denominate  or  concrete  unit,  is  one  thing 
named  or  denominated  ;  as,  one  apple,  one  peach, 
one  pear,  one  horse,  &c. 


*   U2'    Number'    as   such>   has  no  reference 
to  the  particular  things  numbered.   *  But  to  dis- 


CHAP.   II.]  ARITHMETIC ALPHABET.  121 

tinguish  numbers  which  are  applied  to  particular  to  the  things 

units  from  those  which  are  purely  abstract,  we 

call    the    latter  Abstract   or   Simple   Numbers,     simple 

and 

and  the  former  Concrete  or  Denominate  Num-  Denominate, 
bers.      Thus,   fifteen   is  an   abstract   or  simple 
number,  because   the   unit   is   one;  and  fifteen   Examples. 
pounds  is   a    concrete   or   denominate  number, 
because  its  unit,  one  pound,  is  denominated  or 
named. 


ALPHABET WORDS GRAMMAR. 

§  113.  The  term  alphabet,  in  its  most  general    Alphabet, 
sense,  denotes   a  set  of  characters  which  form 
the  elements  of  a  written  language. 

When  any  one  of  these  characters,   or  any     words, 
combination  of  them,  is  used  as  the  sign  of  a 
distinct  notion  or  idea,  it  is  called  a  word ;  and 
the  naming  of  the  characters  of  which  the  word 
is  composed,  is  called  its  spelling. 

Grammar,  as  a  science,  treats  of  the  estab-    Grammar 
lished  connection  between  words  as  the  signs  of 
ideas. 

ARITHMETICAL     ALPHABET. 

§  114.    The  arithmetical  alphabet  consists  of  Arithmetical 

Alphabet. 

ten  characters,  called  figures.     They  are, 

Naught,  One,     Two,  Three,  Four,    Five,     Six,    Seven,  Eight,   Nine, 

0123456789 


122 


MATHEMATICAL     SCIENCE.  [fiOOK  II. 


and  each  may  be  regarded  as  a  word,  since  it 
stands  for  a  distinct  idea. 


WORDS — SPELLING  AND  READING  IN  ADDITION. 

one  cannot  §  115.  The  idea  of  one,  being  elementary,  the 
character  1  which  represents  it,  is  also  element 
ary,  and  hence,  cannot  be  spelled  by  the  other 
characters  of  the  Arithmetical  Alphabet  (§  114). 
But  the  idea  which  is  expressed  by  2  comes  from 

spelling  by  the  addition  of  1  and  1  :  hence,  the  word  repre- 

the 

arithmetical  sented  by  the  character  2,  may  be  spelled  by 
1  and  1.  Thus,  1  and  1  are  2,  is  the  arithmet 
ical  spelling  of  the  word  two. 

Three  is  spelled  thus:    1  and  2  are  3;    and 
also,  2  and  1  are  3. 

:ampies.       Four  ^s  gp^fe^  j  and  3  ^  ^  .  3  and  I  are  4  ; 
2  and  2  are  4. 

Five  is  spelled,  1  and  4  are  5 ;  4  and  1  are  5 ; 
2  and  3  are  5  ;  3  and  2  are  5. 

Six  is  spelled,  1  and  5  are  6 ;  5  and  1  are  6 ; 
2  and  4  are  6;  4  and  2  are  6;  3  and  3  are  6. 

AH  numbers      §  116.    In  a  similar  manner,  any  number  in 

may  be  .  ,  J     ' 

speiied  in  a   arithmetic  may  be  spelled;    and  hence  we  see 

vay-  that  the  process  of  spelling  in  addition  consists 

simply,  in  naming  any  two  elements  which  will 

make  up  the  number.     All  the  numbers  in  ad- 


CHAP. 

ii. 

]                ARITHMETIC  READINGS. 

123 

dition  are  therefore  spelled  with 
The  reading  consists  in  naming 

two  syllables, 
only  the  word 

Reading:  in 

which 

expresses  the  final  idea. 

Thus, 

what  it  con 
sists. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Examples. 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

One 

two      three 

four 

five 

six     seven 

eight 

nine      ten. 

We 

may  now  read 

the  words 

which 

express 

the  first 

hundred  combinations. 

READINGS.                                             Read. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Two,  three, 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

four,  &c. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Three,  four, 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

*«. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Four,  five, 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

&c. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Five,  six,  &c. 

Six,  seven, 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

&c. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Seven,  eight, 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

&c. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Eight,  nine, 

7 

7 

7 

7 

&c. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Nine,  ten,  &.c. 

8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

124 


MATHEMATICAL     SCIENCE.  [l3OOK  II. 


Ten,  eleven, 

&C. 

Eleven, 
twelve,  &c. 

Example  for 
reading  in 
Addition. 

123456789     10 
9999999999 

123456789     10 
10     10     10     10     10     10     10     10     10     10 

§  117.  In  this  example,  beginning 
at  the  right  hand,  we  say,  8,  17,  18, 
26  :  setting  down  the  6  and  carry 
ing  the  2,  we  say,  8,  13,  20,  22,  29  : 

878 
421 
679 
354 
764 

setting   down   the  9  and   carrying 
the  2,  we  say,  9,   12,   18,   22,  30: 

3096 

and  setting  down  the  30,  we  have  the  entire  sum 

AII  examples  3096.    All  the  examples  in  addition  may  be  done 

so  solved. 

in  a  similar  manner. 


Advantages 
of  reading. 

1st.  stated. 


2d.  stated. 


§  118.  The  advantages  of  this  method  of  read 
ing  over  spelling  are  very  great. 

1st.  The  mind  acquires  ideas  more  readily 
through  the  eye  than  through  either  of  the  other 
senses.  Hence,  if  the  mind  be  taught  to  appre 
hend  the  result  of  a  combination,  by  merely  see 
ing  its  elements,  the  process  of  arriving  at  it  is 
much  shorter  than  when  those  elements  are  pre 
sented  through  the  instrumentality  of  sound. 
Thus,  to  see  4  and  4,  and  think  8,  is  a  very  dif 
ferent  thing  from  saying,  four  and  four  are  eight. 

2d.  The  mind  operates  with  greater  rapidity 
and  certainty,  the  nearer  it  is  brought  to  the 


CHAP.   II.]  ARITHMETIC WORDS.  125 

ideas  which  it  is  to  apprehend  and  combine. 
Therefore,  all  unnecessary  words  load  it  and 
impede  its  operations.  Hence,  to  spell  when 
we  can  read,  is  to  fill  the  mind  with  words 
and  sounds,  instead  of  ideas. 

3d.  All  the  operations  of  arithmetic,  beyond  3d.  stated. 
the  elementary  combinations,  are  performed  on 
paper ;  and  if  rapidly  and  accurately  done,  must 
be  done  through  the  eye  and  by  reading.  Hence 
the  great  importance  of  beginning  early  with  a 
method  which  must  be  acquired  before  any  con 
siderable  skill  can  be  attained  in  the  use  of 
figures. 

§  119.  It  must  not  be  supposed  that  the  read-     Reading 

comes  after 

ing  can  be  accomplished  until  the  spelling  has     spelling. 
first  been  learned. 

In  our  common  language,  we  first  learn  the    same  as  in 

our  common 

alphabet,  then  we  pronounce   each  letter  in  a    language. 
word,  and  finally,  we  pronounce  the  word.     We 
should  do  the  same  in  the  arithmetical  reading. 


WORDS SPELLING  AND  READING  IN  SUBTRACTION. 

§  120.  The  processes  of  spelling  and  reading  same  princi 
ple  applied 
which  we  have   explained   in   the   addition   of    m  subtree- 

numbers,  may,  with  slight  modifications,  be  ap 
plied  in  subtraction.    Thus,  if  we  are  to  subtract 


126 

MATHEMATICAL 

SCIENCE. 

[COOK  ii. 

2  from  5,  we  say,  ordinarily,  2  from  5  leaves  3  ; 
or  2  from  5  three  remains.      Now,  the  word, 
three,  is  suggested  by  the  relation  in  which  2 
and  5  stand  to  each  other,  and  this  word  may  be 
Readings  in  read  at  once.     Hence,  the  reading,  in  subtrac- 

Subtraction        .                            7                           ,-1                j        -i  •    7 

explained.    tlon>  ls  simply  naming  the  word,  which  expresses 
the  difference  between  the  subtrahend  and  min 
uend.     Thus,  we  may  read  each  word  of  the 
following  one  hundred  combinations. 

READINGS. 

One  from 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

one,  &c. 

1 

1 

1 

1 

1 

1 

1 

1 

1 

1 

Two  from 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

two,  &c. 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

Three  from 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

three,  &c. 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

Four  from 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

four,  &c. 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

Five  from 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

five,  &c. 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

i 

Six  from  six, 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

&c. 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

Seven  from 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

eeven,  &c. 

7 

7 

7 

7 

CHAP.   II.]  ARITHMETIC SPELLING.  127 

8  9     10     11      12     13     14     15     16     17  Eight  from 
8888888888  eight,  &c. 

9  10       11        12       13       14       15       16       17       18  Ninefrom 

9999999999  nine'&c- 


10       11        12       13       14       15       16       17       18       19          Ten  from  ten, 

10     10     10     10     10     10     10     10     10     10  &c* 


§  121.  It  should  be  remarked,  that  in  subtrac 
tion,  as  well  as  in  addition,  the  spelling  of  the  spelling  pre- 

.  cedes  reading 

words   must  necessarily  precede  their  reading.   msubtrac- 
The  spelling  consists  in  naming  the  figures  with 
which  the  operation  is  performed,  the  steps  of 
the  operation,  and  the  final  result.     The  reading    Reading, 
consists  in  naming  the  final  result  only. 


SPELLING   AND   READING   IN    MULTIPLICATION. 

§  122.  Spelling  in  multiplication  is  similar  to   spelling  in 

.  Multiplica- 

the  corresponding  process  in  addition  or  subtrac-       uon. 
tion.      It   is   simply  naming   the   two   elements 
which  produce  the  product ;  whilst  the  reading    Reading, 
consists   in   naming   only  the  word  which   ex 
presses  the  final  result. 

In  multiplying  each  number  from  1  to  10  by  Examples  in 
2,  we  usually  say,  two  times  1  are  2 ;  two  times 
2  are  4 ;  two  times  3  are  6 ;  two  times  4  are  8  ; 
two  times  5  are  10;  two  times  6  are  12;  two 


128  MATHEMATICAL     SCIENCE.  [fiOOK   II. 

times  7  are  14;  two  times  8  are  16;  two  times 
in  reading.   9  are  18 ;  two  times  10  are  20.     Whereas,  we 
should  merely  read,  and  say,  2,  4,  6,  8,  10,  12, 
14,  16,  18,  20. 

In  a  similar  manner  we  read  the  entire  mul 
tiplication  table. 

READINGS. 
Onceoneis          12       11       10       987654321 


Twotimesi       12     1 1     10     9     8     7     6     5     4     3     2     1 

are  2,  &c. 


Threetimesl         12       1 1       10      9      8      7       6       5      4       3       2       1 

are  3,  &c.  o 


Fourtimesl  12       1  1       10      9       8       7       6       5       4       3       2       1 

are  4,  &c.  A 

Fivetimesl  12       11       10      9      8      7      6       5      4      3      2       1 

are  5,  &c. 


12     11     10     9     8     7     6     5     4     3     2     1 

are  six,  &c.  r* 


Seven  timea          121110987654321 

1  are  7,  &c.  1 


12     11     10     987654321 

are8,&c. 


CHAP.   II.]  ARITHMETIC READING. 


12        11        10       9       8       7       6       5       4       3       2       1  Nine  timos  1 

are  9,  &c. 


12        11        10       987654321  Ten  times  I 

JQ  are  10,  &c. 


12       11        10       987654321  Eleven  times 

1  -i  1  are  11,  <fcc. 


12       11        10       987654321  Twelve  times 

lareJ2,&c. 


SPELLING    AND    READING    IN    DIVISION. 

§  123.  In  all  the  cases  of  short  division,  the  in  short  Dh-i- 

.          ft  s'°n' we  may 

quotient  may  be  read  immediately  without  nam-       read: 

ing  the  process  by  which  it  is  obtained.  Thus, 
in  dividing  the  following  numbers  by  2,  we 
merely  read  the  words  below. 

2)4       6       8       10       12       16       18       22 

two     three     four        five          six         eight       nine       eleven. 

In  a  similar  manner,  all  the  words,  expressing   inaii  cases, 
the  results  in  short  division,  may  be  read. 

READINGS. 

2)2      4      6      8    10    12    14    16    18    20    22    24      Twoins, 

once,  &c. 

3)3      6      9    12    15    18    21   24    27    30    33    36     Three  in  3, 

once,  &c. 

4)4      8    12    16    20    24    28    32    36    40    44    48      Fourin4, 

once,  fcc. 


130 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


Five  in  5,   5)5  10  15  20  25  30  35  40  45  50  55  60 

once,  &.c. 

Fix  in  6,   6)6  12  18  24  30  36  42  48  54  60  66  72 

once,  &c. 

Feven  in  7,      7)7    14    21    28    35    42    49    56    63    70    77    84 

once,  &c. 

Eight  in  8,      8)8    16    24    32    40    48    56    64    72    80    88    96 

once,  &.c. 

Nine  in  9,      9)  9  18  27  36  45  54  63  72     81^    90 99_108 

once,  &c. 

Ten  in  10,     10)lQ  20  30  40  50  60  70  80     90  100  110  120 

once,  &c. 


Eleven  in  ii,  11)11  22  33  44  55  66  77  88     99  110  121   132 

once,  &c. 

Twelve  in  12,  12)12  24  36  48  60  72  84  96  108  120  132  144 

once,  &c. 

.-      r- 

TJNITS    INCREASING   BY    THE    SCALE    OF    TENS. 

The  idea  of  a      §  124.  The  idea  of  a  particular  number  is  ne- 

particular 

number  is    ccssanly  complex ;  for,  the  mind  naturally  asks  : 
1st.  What  is  the  unit  or  basis  of  the  number? 
and, 

2d.   How   many   times   is   the   unit   or   basis 
taken  ? 


What  a  fig-       §  125.  A  figure  indicates  how  many  times  a 

ure  indicates. 

unit  is  taken.  Each  of  the  ten  figures,  however 
written,  or  however  placed,  always  expresses  as 
many  units  as  its  name  imports,  and  no  more ; 
nor  does  the  j%wre  itself  at  all  indicate  the  kind 


CHAP.  II.]     ARITHMETIC  SCALE     OF     TENS. 

] 
131 

of  unit.    Still,  every  number  expressed  by  one  or 
more  figures,  has  for  its  basis  either  the  abstract 
unit  one,  or  a  denominate  unit.*     If  a  denomi 
nate  unit,  its  value  or  kind  is  pointed  out  either 
by  our  common  language,  or  as  we  shall  present 
ly  see,  by  the  place  where  the  figure  is  written. 
The  number  of  units  which  may  be  expressed 
by  either  of  the  ten  figures,  is  indicated  by  the 
name  of  the  figure.     If  the  figure  stands  alone, 
and  the  unit  is  not  denominated,  the  basis  of  the 
number  is  the  abstract  unit  1. 

§  126.  If  we  write  0  on  the  right  of  j 

L                                                                                                                                      *^» 

1,  we  have     ) 
which  is  read  ONE  ten.     Here  1  still  expresses 
ONE,  but  it  is  ONE  ten  ;  that  is,  a  unit  ten  times 
as  great  as  the  unit  1  ;  and  this  is  called  a  unit 
of  the  second  order. 
Again  ;  if  we  write  two  O's  on  the  ) 

Number  has 
one  for  its 
basis. 

Number  ex 
pressed  by  a 
single  figure. 

How  ten  ia 
written. 

Unit  of  the 
second  order. 

How  to  write 
one  hundred. 

A  unit  of  the 
third  order. 

Laws  —  when 
figures  are 
written  by 
the  side  of 
each  other. 

which  is  read  ONE  hundred.     Here  again,  1  still 
expresses  ONE,  but  it  is  ONE  hundred  ;  that  is,  a 
unit  ten  times  as  great  as  the  unit  ONE  ten,  and 
a  hundred  times  as  great  as  the  unit  1. 

§  127.   If  three   1's  are  written  by  j 
Sill, 

V  „  v  ,        

*  Section  111. 

132  MATHEMATICAL     SCIENCE.  [fiOOK  II. 

the  ideas,  expressed  in  our  common  language, 
are  these  : 

First.  1st.  That  the  1  on  the  right,  will  either  express 

a  single  thing  denominated,  or  the  abstract  unit 
one. 
second.          2d.   That  the  1  next  to  the  left  expresses  1  ten  ; 

that  is,  a  unit  ten  times  as  great  as  the  first. 
Third.  3d.  That  the  1  still  further  to  the  left  expresses 

1  hundred  ;  that  is,  a  unit  ten  times  as  great  as 
the  second,  and  one  hundred  times  as  great  as  the 
first  ;  and  similarly  if  there  were  other  places. 
what  the        When  figures  are  thus  written  by  the  side  of 
eacn  other,  the  arithmetical  language  establishes 
a  relation  between  the  units  of  their  places  :  that 
^       is,  the  unit  of  each  place,  as  we  pass  from  the 
right  hand  towards  the  left,  increases  according 
to  the  scale  of  tens.     Therefore,  by  a  law  of  the 
arithmetical  language,  the  place  of  a  figure  fixes 
its  unit. 
scale  for        If,  then,  we  write   a  row  of  O's  as   a  scale 

Numeration.      , 

thus  : 


a         3 
s  rf      a 


f3     3   ra      3 
.q  -S  ,0    j 


The  units  of  000,    000,    000,    000 

place  deter- 

the  unit  of  each  place  is  determined,  as  well 


CHAP.   II.]      ARITHMETIC SCALE     OF     TENS.  133 


as  the  law  of  change  in  passing  from  one  place 

to  another.     If  then,  it  were  required  to  express     HOW  any 

.  number  of 

a  given  number  01  units,  ot  any  order,  we  first  units  may  be 
select  from  the  arithmetical  alphabet  the  char 
acter  which  designates  the   number,  and   then 
write  it  in  the  place  corresponding  to  the  order. 
Thus,  to  express  three  millions,  we  write 

3000000 ; 
and  similarly  for  all  numbers. 

§  128.   It  should  be   observed,  that   a   figure  A  figure  has 
being   a   character  which  represents  value,  can       itsein 
have  no  value  in  and  of  itself.     The  number  of 
things,  which  any  figure  expresses,  is  determined 
by  its  name,  as  given  in  the  arithmetical  alpha 
bet.     The  kind  of  thing,  or  unit  of  the  figure,  is  HOW  the  unit 

~        ,      .  ,          ,  .  A,  c         i  isdeter- 

fixed  either  by  naming  it,  as  in  the  case  of  a  de-     mined. 
nominate  number,  or   by  the  place   which  the 
figure  occupies,  when  written  by  the  side  of  or  . 
over*  other  figures. 

The  phrase  "  local  value  of  a  figure,"  so  long  Figure,  has 

. ,,         .  no  local 

in  use,  is,  therefore,  without  signification  when      value> 

applied   to    a   figure :    the   term   "  local   value," 

being  applicable  to  the  unit  of  the  place,  and  Termappii- 

_  1-1  i          i  cable  to 

not  to  the  figure  which  occupies  the  place.  unit  ofplace, 

§  129.  Federal  Money  affords  an  example  of  a     Federal 

Money. 

*  Section  199. 


134  MATHEMATICAL     SCIENCE.  [BOOK  II. 


itsdenomina-  series  of  denominate  units,  increasing  according 


to  the  scale  of  tens  :  thus, 


4      c       of  +?      „ 

be  a    g    g  id 
ca   o  .a   ®  a 

P3    ft    fi    U    S 

11111 

HOW  read,  may  be  read  11  thousand  1  hundred  and  11 
mills  ;  or,  1111  cents  and  1  mill  ;  or,  111  dimes 
]  cent  and  1  mill;  or,  11  dollars  1  dime  1  cent 
and  1  mill;  or,  1  eagle  1  dollar  1  dime  1  cent 
and  1  mill.  Thus,  we  may  read  the  number 


with  either  of  its  units  as  a  basis,  or  we  may 
name  them  all  :  thus,  1  eagle,  1  dollar,  1  dime, 
1  cent,  1  mill.  Generally,  in  Federal  Money, 
we  read  in  the  denominations  of  dollars,  cents, 
and  mills;  and  should  say,  11  dollars  11  cents 
and  1  mill. 

Examples  in       §  130.  Examples  in  reading  figures  :  — 

ist.  Example.      If  we  have  the  figures     -     -     -     -  89 

we  may  read  them  by  their  smallest 

unit,  and  say  eighty-nine  ;   or,  we  may  say  8 

tens  and  9  units. 
ad.  Example.      Again,  the  figures    ......         567 

may  be  read   by  the    smallest   unit; 

viz.  five  hundred  and  sixty-seven  ;  or  we   may 

say,  56  tens  and  7  units  ;  or,  5  hundreds  6  tens 

and  7  units. 
3d  Example.      Again,  the  number  expressed  by     -     74896 


CHAP.   II.]       ARITHMETIC VARYING     SCALES.  135 

may  be  read,  seventy-four  thousand  eight  hun-  various  read 
ings  of  a 
dred  and  ninety-six.     Or,  it  may  be  read,  7489    number. 

tens  and  6  units ;  or,  748  hundreds  9  tens  and 
6  units ;  or,  74  thousands  8  hundreds  9  tens 
and  6  units  ;  or,  7  ten  thousands  4  thousands  8 
hundreds  9  tens  and  6  units ;  and  we  may  read 
in  a  similar  way  all  other  numbers. 

Although  we  should  teach  all  the  correct  read-     The  best 
ings  of  a  number,  we  should  not  fail  to  remark     reading. 
that  it  is  generally  most  convenient  in  practice 
to  read  by  the  lowest  unit  of  a  number.     Thus, 
in  the  numeration  table,  we  read  each  period  by  Each  period 
the  lowest  u 
the  number 


T         TI  -.  read  by  its 

the  lowest  unit  of  that  period,     r  or  example,  in  lowest  im[L 


874,967,847,047,  Example. 

we  read  874  billions  967  millions  847  thousands 

and  47. 

UNITS    INCREASING    ACCORDING    TO    VARYING    SCALES. 

§  131.  If  we  write  the  well-known  signs  of    Methods  of 

writing  fig- 

the  English  money,  and  place  1  under  each  de-   uri,s  havm., 

i      11    i  "       different 

nomination,  we  shall  have  denominate 

units. 
£.     *.      d.     f. 

1111 

Now,  the  signs  £.  s.  d.  and/,  fix  the  value  of     HOW  the 

value  of  each 

the  unit  1  in  each  denomination;  and  they  also  unit  is  fixed. 


136  MATHEMATICAL     SCIENCE.  [BOOK   IJ. 


what  the    determine  the  relations  which  subsist  between 

liinimnge 
expresses. 


the   different   units.     For   example,  this    simple 


language  expresses  these  ideas  : 

The  units  of       1st.  That  the  unit  of  the  right-hand  place  is 
the  places.    ^  farthing— of  the  place  next  to  the  left,  1  penny 

— of  the  next  place,  1  shilling — of  the  next  place, 

1  pound  ;  and 
HOW  the        2d.  That  4  units  of  the  lowest  denomination 

make  one  unit  of  the  next  higher;   12  of  the 

second,  one  of  the  third ;  and  20  of   the  third, 


increase. 


one  of  the  fourth. 
The  units  m       If  we  take  the  denominate  numbers  of  the 

Avoirdupois      .         .     ,  .  .    , 

weight.     Avoirdupois  weight,  we  have 

Ton.  cwt.   qr.    Ib.     oz.    dr. 

111111; 

changes  in   in    which   the   units    increase   in   the   following 

the  value  of  ,  , 

the  units.  manner :  viz.  the  second  unit,  counting  from 
the  right,  is  sixteen  times  as  great  as  the  first ; 
the  third,  sixteen  times  as  great  as  the  second ; 
the  fourth,  twenty-five  times  as  great  as  the 
third ;  the  fifth,  four  times  as  great  as  the  fourth ; 
and  the  sixth,  twenty  times  as  great  as  the  fifth, 
uowthe scale  The  scale,  therefore,  for  this  class  of  denominate 

varies. 

numbers  varies  according  to  the  above  laws. 
A  different        If   we    take    any    other   class    of  denominate 

scale  for  each 

system,  numbers,  as  the  Troy  weight,  or  any  of  the 
systems  of  measures,  we  shall  have  different 
scales  for  the  formation  of  the  different  units. 


CHAP.   II.]        ARITHMETIC INTEGER     UNITS.  137 


But  in   all   the   formations,   we  shall   recognise  The  method 
the  application  of  the  same  general  principles.       the  °c.i}^  the 
There  are,  therefore,  two  general  methods  of  ^jj^J111 
forming  the  different  systems  of  integer  num- 

i  r  -  mi          e  •  •       Two  systems 

oers  irom  the  unit  one.  Ihe  nrst  consists  m  Of  forming 
preserving  a  constant  law  of  relation  between  ln  1^° 
the  different  unities  ;  viz.  that  their  values  shall 

First  system. 

change    according   to  the  scale  of  tens.     This 
gives  the  system  of  common  numbers. 

The  second  method  consists  in  the  application   second  Bys- 
of  known,  though  varying  laws  of  change  in  the 
unities.     These  changes  in  the  unities  produce  change  in  the 
the  entire  system  of  denominate  numbers,  each  fj^inj'the 
class  of  which  has  its  appropriate  scale,  and  the 
changes  among  the  units  of  the  same  class  are 
indicated  by  the  different  degrees  of  its  scale. 


INTEGER    UNITS    OF    ARITHMETIC. 

§  132.  There  are  four  principal  classes  of  units  Four  classes 

...  of  units. 

in  arithmetic : 

1st.  Abstract,  or  simple  units  ;  1st.  class. 

2d.  Units  of  Currency  ;  2d.  class. 

3d.  Units  of  Weight;  and  3d. class. 

4th.  Units  of  Measure.  4th.  clasSi 

First  among  the   Units  of  arithmetic   stands 
the  simple  or  abstract  unit  1.     This  is  the  basis  Abstract  unit 
of  all  simple  numbers,  and  becomes   the  basis,  OI 


138 


MATHEMATICAL     SCIENCE.  [BOOK   II. 


Thebasi9of  also,  of  all  denominate  numbers,  by  merely  na- 

denominate 
numbers ; 


""          ming,   in    succession,    the    particular    things    to 


which  it  is  applied. 

AISO,  the  ba-  It  is  also  the  basis  of  all  fractions.  Merely  as 
the  unit  1,  it  is  a  whole  which  may  be  divided 
according  to  any  law,  forming  every  variety  of 
inate-  fraction  ;  and  if  we  apply  it  to  a  particular  thing, 
the  fraction  becomes  denominate,  and  we  have 
expressions  for  all  conceivable  parts  of  that  thing. 

§  133.  It    has  been    remarked*   that   we  can 

Must  appre-  form  no  distinct  apprehension  of  a  number,  un 
bend  the 
unit.       til  we  have  a  clear  notion  of  its  unit,  and  the 

number  of  times  the  unit  is  taken.     The  unit  is 
the  great   basis.      The   utmost    care,   therefore, 
Let  its  nature  should   be  taken  to  impress    on   the   minds    of 
fuiiy  explain-  learners,  a  clear  and  distinct  idea  of  the  actual 
value  of  the  unit  of  every  number  with  which 
they  have  to  do.     If  it  be  a  number  expressing 
\_  currency,  one  or  more  of  the  coins  should  be 


pressing  cur-  exhibited,  and  the  value  dwelt  upon:  after  which, 

rency. 

distinct  notions   of  the  other  units  can  be  ac 
quired  by  comparison. 

Exhibit  the        ^  ^e  number  be   one  of  weight,  some  unit 
11  be   should  be  exhibited,  as  one  pound,  or  one  ounce, 
and  an  idea  of  its  weight  acquired  by  actually 

*  Section  110. 


CHAP.  II.]          ARITHMETIC  -  FEDERAL     MONEY.  139 

lifting  it.     This  is  the  only  way  in  which  we 
can  learn  the  true  signification  of  the  terms. 

If  the    number    be    one    of  measure,   either  And  also,  if  it 
linear,  superficial,  liquid,  or  solid,  its  unit  should    measure. 
also  be  exhibited,  and  the   signification    of  the 
term  expressing  it,  learned  in  the  only  way  in 
which  it  can  be  learned,  through  the  senses,  and 
by  the  aid  of  a  sensible  object. 

FEDERAL      MONEY. 

§  134.  The  currency  of  the  United  States  is  currency  of 
called  Federal  Money.     Its  units  are  all  denomi 
nate,  being   1   mill,    1  cent,  1  dime,  1   dollar,   ] 

eagle.     The  law  of  change,  in  passing  from  one  Law  of 

change  in  the 

unit  to  another,  is  according  to  the  scale  of  tens, 


Hence,  this  system  of  numbers  may  be  treated, 

J  J  How  these 

in   all  respects,  as  simple  numbers  ;   and  indeed  numbers  may 

be  treated. 

they  are  such,  with  the  single  exception  that 
their  units  have  different  names. 

They  are  generally  read  in  the  units  of  dollars,    HOW  geu- 

.  ,  r  erallyread. 

cents,  and  mills  —  a  comma  being  placed  after 
the  figure  denoting  dollars.  Thus, 

$  864,849  Example. 

is  read  eight  hundred  and  sixty-four  dollars, 
eighty-four  cents,  and  nine  mills  ;  and  if  there 
were  a  figure  after  the  9,  it  would  be  read  in 
decimals  of  the  mill.  The  number  may,  how- 


140  MATHEMATICAL     SCIENCE.  [BOOK  II. 


The  number  ever,  be   read   in    any  other   unit ;    as,  864849 

read  in 
various  ways. 


mills  ;    or,  86484  cents   and  9   mills  ;    or,   8648 


dimes,  4  cents,  and  9  mills ;  or,  86  eagles,  4  dol 
lars,  84  cents,  and  9  mills;  and  there  are  yet 
several  other  readings. 


ENGLISH     MONEY. 

sterling  MO-  §  135.  The  units  of  English,  or  Sterling  Mo 
ney,  are  1  farthing,  1  penny,  1  shilling,  and  1 
pound. 

scale  of  the       The  scale  of  this  class  of  numbers  is  a  varying 

scale.     Its  degrees,  in  passing  from  the  unit  of 

the  lowest  denomination  to  the  highest,  are  four, 

HOW  it     twelve,  and  twenty.     For,  four  farthings  make 

changes.  ,  .„. 

one  penny,  twelve  pence  one  shilling,  and  twenty 
shillings  one  pound. 


AVOIRDUPOIS     WEIGHT. 

units  in         §  136.  The  units  of  the  Avoirdupois  Weight 

Avoirdupois. 

are  1  dram,  1  ounce,  1  pound,  1  quarter,  1  hun 
dred-weight,  and  1  ton. 

scale.  The  scale  of  this  class  of  numbers  is  a  vary 

ing  scale.     Its  degrees,  in  passing  from  the  unit 
of  the  lowest  denomination  to  the  highest,  are 
sixteen,  sixteen,  twenty-five,  four,  and    twenty. 
Variation  in  For,    sixteen   drams    make    one    ounce,    sixteen 
>ee?-  ounces  one  pound,  twenty-five  pounds  one  quar- 


CHAP.   II.]  ARITHMETIC UNITS    OF    LENGTH.  141 

ter,  four  quarters  one  hundred,  and  twenty  hun 
dreds  one  ton. 


TROY     WEIGHT. 

§  137.  The  units  of  the  Troy  Weight  are,  1     Unit9  in 

Troy  Weight. 

grain,  1  pennyweight,  1  ounce,  and  1  pound. 

The  scale  is  a  varying  scale,  and  its  degrees,      scale: 
in  passing  from  the  unit  of  the  lowest  denomina-    Its  <\eSTec9f 
tion  to  the  highest,  are  twenty-four,  twenty,  and 
twelve. 

APOTHECARIES'    WEIGHT. 
§  138.   The  units  of  this  weight  are,  1  grain,  1     units  in 

Apothecaries 

scruple,  1  dram,  1  ounce,  and  1  pound.  weight. 

The  scale  is  a  varying  scale.     Its  degrees,  in      Scalo: 
passing  from  the  unit  of  the  lowest  denomina-    it*  degrees 
tion  to  the  highest,  are  twenty,  three,  eight,  and 
twelve. 

UNITS     OF     MEASURE. 

§  139.  There  are  three  units  of  measure,  each   Three  units 

of  measure. 

differing  in  kind  from  the  other  two.  They  are, 
Units  of  Length,  Units  of  Surface,  and  Units  of 
Solidity. 

UN  ITS     OF     LENGTH. 

§  140.  The  unit  of  length  is  used  for  measur-      units  of 

length. 

ing   lines,   either    straight   or   curved.      It   is    a 


142 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


The  stand-   straight  line  of  a  given  length,  and  is  often  called 

ard. 

the  standard  of  the  measurement. 

what  units       The  units  of  length,  generally  used  as  stand- 
are  taken. 

ards,  are  1  inch,  1  foot,  1  yard,  1  rod,  1  furlong, 

and   1   mile.     The  number  of  times  which  the 

idea  of     unit,  used  as  a  standard,  is  taken,  considered  in 

length. 

connection  with  its  value,  gives  the  idea  of  the 
length  of  the  line  measured. 


Units  of 
surface. 


What  the 

unit  of 
surface  is. 


1  square  foot. 


Its  connection 

with  the  unit 

of  length. 


Square  feet 

in  a 
square  yard. 


UNITS     OF     SURFACE. 

§  141.  Units  of  surface  are  used  for  the  meas 
urement  of  the  area  or  contents  of  whatever  has 
the  two  dimensions  of  length  and  breadth.  The 
unit  of  surface  is  a  square  de 
scribed  on  the  unit  of  length 

o 

as  a  side.     Thus,  if  the  unit 
of  length  be  1  foot,  the  corre 
sponding  unit  of  surface  will 
be  1  square  foot;  that  is,  a  square  constructed  on 
1  foot  of  length  as  a  side. 

If  the  linear  unit  be  1  yard, 
the  corresponding  unit  of  sur 
face  will  be  1  square  yard.  It 
will  be  seen  from  the  figure, 
that,  although  the  linear  yard 
contains  the  linear  foot  but 
three  times,  the  square  yard 


1  yard. 


CHAP.   II.]       ARITHMETIC DUODECIMAL     UNITS.  143 


contains  the  square  foot  nine  times.  The  square  square  rod 
rod  or  square  mile  may  also  be  used  as  the  unit  square  mile, 
of  surface. 

The  number  of  times  which  a  surface  contains     Area  or 

.      .  ,    contents  of  a 

its  unit  of  measure,  is  its  area  or  contents  ;  and     8Urface. 
this  number,  taken  in  connection  with  the  value 
of  the  unit,  gives  the  idea  of  its  extent. 

Besides  the  units  of  surface  already  considered, 
there  is  another  kind,  called, 


DUODECIMAL     UNITS. 

§  142.    The   duodecimal   units    are   generally  Duodecimal 
used  in  board  measure,  though  they  may  be  used 
in  all  superficial  measurements,  and  also  in  solid. 

The  square  foot  is  the  basis  of  this  class  of  Their  basis. 
units,  and  the  others  are  deduced  from  it,  by  a 
descending  scale  of  twelve. 


§  143.  It  is  proved  in  Geometry,  that  if  the  what 
number  of  linear  units  in  the  base  of  a  rectan- 
gle  be  multiplied  by  the  number  of  linear  units 
in  the  height,  the  numerical  value  of  the  pro 
duct  will  be  equal  to  the  number  of  superficial 
units  in  the  figure. 

Knowing  this  fact,  we  often  express  it  by  say-  Howitis  ex- 
ing,  that  "feet  multiplied  by  feet  give    square     pre£ 
feet,"  and  "yards  multiplied  by  yards  give  square 


144  MATHEMATICAL     SCIENCE.  [BOOK   II. 


yards."  But  as  feet  cannot  be  taken  feet  times, 
nor  yards  yard  times,  this  language,  rightly  un 
derstood,  is  but  a  concise  form  of  expression  for 
the  principle  stated  above. 

conclusion.  With  this  understanding  of  the  language,  we 
say,  that  1  foot  in  length  multiplied  by  1  foot  in 
height,  gives  a  square  foot ;  and  4  feet  in  length 
multiplied  by  3  feet  in  height,  gives  12  square 
feet. 


Examples  in          §    144.     If    nOW,    1     foot     ill 
the  muIlipH- 

cationoffeet  length  be  multiplied  by  1  inch 

by  feet  and  , 

inches.  =T5  °f  a  foot  in  height,  the 
product  will  be  one-twelfth 
of  a  square  foot ;  that  is,  one- 
twelfth  of  the  first  unit:  if  it 
be  multiplied  by  3  inches,  the  product  will  be 

Generaiiza-   three-twelfths  of  a  square   foot;    and    similarly 
for  a  multiplier  of  any  number  of  inches. 

inches  by         If,   now,  we  multiply   1   inch  by   1   inch,  the 

inches. 

product  may  be  represented  by  1  square  inch : 
iiowtheunits  that  is,  by  one-twelfth  of  the  last  unit.     Hence, 

change,  and 

what  they    tke  units  of  this  measure  decrease  according  to 

the  scale  of  12.     The  units  are, 
First.  1st.  Square  feet— arising  from  multiplying  feet 

by  feet. 

second.         2d.  Twelfths  of  square  feet— arising  from  mul 
tiplying  feet  by  inches. 


CHAP.  II.] 


ARITHMETIC  -  UNITS 


115 


3d.  Twelfths  of  twelfths  —  arising  from  multi-      Third. 
plying  inches  by  inches. 

The  same  remarks  apply  to  the  smaller  di- 
visions  of  the  foot,  according  to  the  scale  of 
twelve. 

The  difficulty  of  computing  in  this  measure    Difficulty. 
arises  from  the  changes  in  the  units. 


UNITS    OF    SOLIDITY. 

§  145.  It  has  already  been  stated,  that  if 
length  be  multiplied  by  breadth,  the  product 
may  be  represented  by  units  of  surface.  It  is 

~  Tiii 

also  proved,  in  Geometry,  that  if  the  length, 
breadth,  and  height  of  any  regular  solid  body, 
of  a  square  form,  be  multiplied  together,  the 
product  may  be  represented  by  solid  units  whose 
number  is  equal  to  this  product.  Each  solid 
unit  is  a  cube  constructed  on  the  linear  unit  as 
an  edge.  Thus,  if  the  linear  unit  be  1  foot,  the 
solid  unit  will  be  1  cubic  or  solid  foot  ;  that  is, 
a  cube  constructed  on  1  foot  as  an  edge  ;  and 
if  it  be  1  yard,  the  unit  will  be  1  solid  yard. 

The  three  units,  viz.  the  unit  of  length,  the 
unit  of  surface,  and  the  unit  of  solidity,  are  es- 
sentially  different  in  kind.  The  first  is  a  line 
of  a  known  length  ;  the  second,  a  square  of  a 
known  side  :  and  the  third,  a  solid,  called  a 

10 


units  of  sou- 
what  is 

proved  in  Go- 

Ometry  in  re- 
g£ 


solid  units. 


Examples. 


The  three 
tiaiiy  duier- 

what  they 


J 


116 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


Generally    cube,  of  a  known  base  and  height.     These  are 
used'       the  units  used  in  all  kinds  of  measurement- 
Duodecimal  excepting   only  the   duodecimal   system,  which 
has  already  been  explained. 


LIQUID     MEASURE. 

units  of  LI-       §  146.   The  units  of  Liquid  Measure   are,   1 
quid  Meas-  .  quart,  1  gallon,  1  barrel,  1   hogs 


head,   1    pipe,   1   tun.     The  scale  is   a  varying 
scale.     Its  degrees,  in  passing  from  the  unit  of 
owitva-    the   lowest   denomination,    are,  four,   two,   four, 
thirty-one  and  a  half,  sixty-three,  two,  and  two. 


DRY    MEASURE. 

Units  of  Dry       §  147.  The  units  of  this  measure  are,  1  pint, 

1  quart,  1  peck,  1  bushel,  and  1  chaldron.     The 

Decrees  of    degrees  of  the  scale,  in  passing  from  units  of  the 

the  scale.  .  . 

lowest  denomination,  are  two,  eight,  lour,  and 
thirty-six. 

TIME. 

Units  of         §  148.  The  units  of   Time   are,    1   second,   1 

Time. 

minute,  1  hour,  1  day,  1  week,  1  month,  1  year, 
Degrees  of    and   1   century.      The  degrees   of  the  scale,   in 

the  scale. 

passing  from  units  of  the  lowest  denomination  to 
the  highest,  are  sixty,  sixty,  twenty-four,  seven, 
four,  twelve,  and  one  hundred. 


CHAP.  II.]  ARITHMETIC ADVANTAGES.  147 


CIRCULAR     MEASURE. 


§  149.  The  units  of  this  measure  are,  1  sec-  units  of  cir 
cular  Meas- 
ond,   1  minute,  1  degree,  1  sign,  1   circle.     The       ure. 

degrees  of  the  scale,  in  passing  from  units  of  the    Degrees  of 

the  Scale. 

lowest  denomination  to  those  of  the  higher,  are 
sixty,  sixty,  thirty,  and  twelve. 


ADVANTAGES    OF    THE    SYSTEM    OF    UNITIES. 

§  150.  It  may  well  be  asked,  if  the  method   Advantages 

of  the  system. 

here  adopted,  of  presenting  the  elementary  prin 
ciples  of  arithmetic,  has  any  advantages  over 
those  now  in  general  use.  It  is  supposed  to  pos 
sess  the  following : 

1st.  The  system  of  unities   teaches   an  exact  1st.  Teaches 
analysis  of  all  numbers,  and  unfolds  to  the  mind  Of  numbers: 
the  different  ways  in  which  they  are  formed  from 
the  unit  one,  as  a  basis. 

2d.  Such  an  analysis  enables  the  mind  to  form  2d.  Points  out 
a  definite  and  distinct  idea  of  every  number,  by    relation.- 
pointing  out  the  relation  between  it  and  the  unit 
from  which  it  was  derived. 

3d.  By  presenting  constantly  to  the  mind  the  3d.  constant- 
idea  of  the  unit  one,  as  the  basis  of  all  numbers,   the  idea  of 
the  mind  is  insensibly  led  to  compare  this  unit      muty> 
with  all  the  numbers  which  flow   from  it,  and 


148 


MATHEMATICAL     SCIENCE 


[BOOK  II. 


then  it  can  the  more  easily  compare  those  num 
bers  with  each  other. 

4th.  Explains  4th.  It  affords  a  more  satisfactory  analysis, 
and  a  better  understanding  of  the  four  ground 
ruies>  and  indeed  of  all  the  operations  of  arith 
metic,  than  any  other  method  of  presenting  the 
subject. 


ground 
rules. 


FOUR      GROUND     RULES. 


system 


Examples. 


§  151.  Let  us  take  the  two  following  examples 
m  Addition,  the  one  in  simple  and  the  other  in 
denominate  numbers,  and  then  analyze  the  pro 
cess  of  finding  the  sum  in  each. 


SIMPLE  NUMBERS. 

"874198 
36984 

3641 


914823 


DENOMINATE  NUMBERS 

cwt.  qr.  lb.  oz.  dr. 
3  3  24  15  14 
6  3  23  14  8 

10     3     23      14       6 


Process  of 


But  ouo 


In  both  examples  we  begin  by  adding  the  units 
°f  t^ie  l°west  denomination,  and  then,  we  divide 
their  sum  by  so  many  as  make  one  of  the  denomi 
nation  next  higher.  We  then  set  down  the 
remainder,  and  add  the  quotient  to  the  units 
of  that  denomination.  Having  done  this,  we 
apply  a  similar  process  to  all  the  other  denomina- 
tions  —  the  principle  being  precisely  the  same  in 
both  examples.  We  see,  in  these  examples,  an 


CHAP.  II.]          ARITHMETIC SUBTRACTION.  149 


illustration    of  a   general   principle  of  addition,   units  of  the 
viz.  that  units  of  the  same  kind  are  always  added      unite. 
together. 


§  152.  Let  us  take  two  similar  examples   in      system 

0    -,  .  applied  in 

Subtraction.  subtraction. 


SIMPLE  NUMBERS.  DENOMINATE  NUMBERS. 

8403  £       ,       A    far. 

3298  6972 

5105  3    10      8     4 


2    18    10     2 


In  both  examples  we  begin  with  the  units  of  The  method 

of  performing 

the  lowest  denomination,  and  as  the  number  in  the  examples. 
the  subtrahend  is  greater  than  in  the  place  di 
rectly  above,  we  suppose  so  many  to  bte  added 
in  the  minuend  as  make  one  unit  of  the  next 
higher  denomination.  We  then  make  the  sub 
traction,  and  add  1  to  the  units  of  the  subtrahend 
next  higher,  and  proceed  in  a  similar  manner, 
through  all  the  denominations.  It  is  plain  that 
the  principle  employed  is  the  same  in  both  exam-  Principle  the 

,  _  same  for 

pies.     Also,  that  units  ol  any  denomination  in  aii  examples. 
the  subtrahend  are  taken  from  those  of  the  same 
denomination  in  the  minuend. 


§  153.  Let  us  now  take  similar  examples  in 

»«•!•!•  • 

Multiplication. 


150 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


SIMPLE  NUMBERS.  DENOMINATE  NUMBERS. 

Examples.  87464  ib    §     3     9      gr. 

5  9    7  6    2    15 


437320  5 

48    3  2    1     15 


Method  of        In  these  examples  we  see,  that  we  multiply,  in 
performing   successiorlj  each  order  of  units  in  the  multipli- 

the 

examples.  cand  by  the  multiplier,  and  that  we  carry  from 
one  product  to  another,  one  for  every  so  many  as 
make  one  unit  of  the  next  higher  denomination. 

The  princi 
ple  the  same  The    principle   of  the    process   is   therefore   the 

for  all 

examples,    same  in  both  examples. 

§  154.  Finally,  let  us  take  two  similar  exam 
ples  in  Division. 

SIMPLE  NUMBERS.  DENOMINATE  NUMBERS. 

3)874911  £      s.      d    far, 
291637                           3)8       4      2       1 
2     14      8      3 

Principles  in-  We  begin,  in  both  examples,  by  dividing  the 
voived:  units  of  the  highest  denomination.  The  unit  of 
the  quotient  figure  is  the  same  as  that  of  the 
dividend.  We  write  this  figure  in  its  place,  and 
then  reduce  the  remainder  to  units  of  the  next 
lower  denomination.  We  then  add  in  that  de- 

The  same  as 

in  the      nomination,  and   continue  the  division  through 

other  rules.        n      i          i  •          •  •        •    i 

ail  the  denominations  to  the  last — the  principle 
being  precisely  the  same  in  both  examples. 


CHAP.   II.]  ARITHMETIC FRACTIONS.  151 


SECTION   II. 


FRACTIONAL      UNITS. 


FRACTIONAL    UNITS. SCALE    OF    TENS. 

§  155.  IF  the  unit  1  be  divided  into  ten  equal  Fraction  one- 
parts,  each  part  is  called  one  tenth.     If  one  of     Jjj"^. 
these  tenths   be   divided  into   ten    equal   parts, 
each  part  is  called  one  hundredth.     If  one  of  the   hundredth ; 
hundredths  be  divided  into  ten  equal  parts,  each        One 
part  is  called  one  thousandth  ;  and  corresponding   thousandth- 
names  are  given  to  similar  parts,  how  far  soever   Generaiiza- 
the  divisions  may  be  carried. 

Now,  although   the    tenths  which   arise  from  Fractions  are 
dividing  the  unit  1,  are  but   equal  parts  of  1,      whole 

things. 

they  are,  nevertheless,  WHOLE  tenths,  and  in  this 
light  may  be  regarded  as  units. 

To  avoid  confusion,  in  the  use  of  terms,  we  Fractional 
shall  call  every  equal  part  of  1  a  fractional  unit. 
Hence,  tenths,  hundredths,  thousandths,  tenths 
of  thousandths,  &c.,  are  fractional  units,  each 
having  a  fixed  relation  to  the  unit  1,  from  which 
it  was  derived. 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


Fractional         §  156.  Adopting   a   similar  language    to    that 

units  of  the  .  ill  i         r 

first  order;    used  in  integer  numbers,  we  call  the  tenths,  irac- 


units  of  the 


tional  units  of  the  first  order  >'  the  hundredths, 
fractional  units  of  the  second  order  ;  the  thou 
sandths,  fractional  units  of  the  third  order  ;  and 
so  on  for  the  subsequent  divisions. 

Language  for      Is  tnere  an7  arithmetical  language  by  which 
fractional    these  fractional  units  may  be  expressed  ?     The 

units.  J 

decimal  point,  which  is  merely  a  dot,  or  period, 
whatitfixes.  indicates  the  division  of  the  unit  1,  according  to 

the  scale  of  tens.  By  the  arithmetical  language, 
Names  of  the  the  unit  of  the  place  next  the  point,  on  the  right, 

places. 

is  1  tenth  ;    that  of  the  second   place,  1    hun 
dredth  ;  that  of  the  third,  1  thousandth  ;  that  of 
the  fourth,    1    ten   thousandth;    and   so   on   for 
places  still  to  the  right. 
sc316-          The  scale  for  decimals,  therefore,  is 

.000000000,  &c.  ; 

in  which  the   unit   of  each  place  is  known  as 

soon  as  we  have  learned  the  signification  of  the 

language. 

If,  therefore,  we  wish  to  express  any  of  the 

parts  into  which  the  unit  1  may  be  divided,  ac- 
Any  decimal  cording  to  the  scale  of  tens,  we  have  simply  to 
LTx^reS  Select  from  the  alPhabet>  the  figure  that  will 
oy  this  scale.  express  the  number  of  parts,  and  then  write  it  in 


CHAP.   II.]  ARITHMETIC FRACTIONS. 


153 


the  place  corresponding  to  the  order  of  the  unit,   where  any 
Thus,  to  express  four  tenths,  three  thousandths,     written. 
eight   ten-thousandths,    and    six    millionths,   we 
write 

.403806  ;  Example. 

and   similarly,   for   any   decimal   which   can  be 
named. 

§  157.  It  should  be  observed  that  while   the 
units  of  place  decrease,  according  to  the  scale  of 

tens,  from  left  to  right,  they  increase  according  The  units  in 
crease  from 
to  the  same  scale,  from  right  to  left.     This  is  the  right  to  left. 

same  law  of  increase  as  that  which  connects  the 

units  of  place  in  simple  numbers.     Hence,  simple  consequence. 

numbers  and  decimals  being  formed  according  to 

the  same  law,  may  be  written  by  the  side  of  each 

other  and  treated  as  a  single  number,  by  merely 

preserving    the    separating    or    decimal     point. 

Thus,  8974  and  .67046  may  be  written 


8974.67046 ; 

since  ten  units,  in  the  place  of  tenths,  make  the 
unit  one  in  the  place  next  to  the  left. 


Example. 


FRACTIONAL    UNITS    IN    GENERAL. 


§  158.  If  the  unit  1  be  divided  into  two  equal      Ahai£ 
parts,  each  part  is  called  a  half.     If  it  be  divided 


A  fourth. 
A  fifth. 

Generally. 


These  units 

are  whole 

things. 


Examples. 


MATHEMATICAL     SCIENCE.  [BOOK  II. 

into  three  equal  parts,  each  part  is  called  a  third : 
if  it  be  divided  into  four  equal  parts,  each  part  is 
called  a  fourth  :  if  into  five  equal  parts,  each 
part  is  called  a  fifth ;  and  if  into  any  number  of 
equal  parts,  a  name  is  given  corresponding  to  the 
number  of  parts. 

Now,  although  these  halves,  thirds,  fourths, 
fifths,  &c.,  are  each  but  parts  of  the  unit  1,  they 
are,  nevertheless,  in  themselves,  whole  things. 
That  is,  a  half  is  a  whole  half;  a  third,  a  whole 
third;  a  fourth,  a  whole  fourth;  and  the  same 
for  any  other  equal  part  of  1.  In  this  sense, 
therefore,  they  are  units,  and  we  call  them  frac 
tional  units.  Each  is  an  exact  part  of  the  unit 
1,  and  has  a  fixed  relation  to  it. 

§  159.  Is  there  any  arithmetical  language  by 
which  these  fractional  units  can  be  expressed  ? 
Language  for      The  bar,  written  at  the  right,  is  the 

fractions. 

sign  which  denotes  the  division  of  the 
unit  1  into  any  number  of  equal  parts. 
TO  express       If  we  wish  to  express  the  number  of  equal 

the  number 

of  equal     parts  into  which  it  is  divided,  as  9,  for         __ 
example,  we  simply  write  the  9  under 
the  bar,  and  then  the  phrase  means,  that  some 
thing  regarded  as  a  whole,  has  been  divided  into 
9  equal  parts. 


Have  a  rela 
tion  to  unity. 


CHAP.   II. J  ARITHMETIC FRACTIONS.  155 


To  show  how 

many  are 

taken. 


If,  now,  we  wish  to  express  any 
number  of  these  fractional  units,  as  7, 
for  example,  we  place  the  7  above  the 
line,  and  read,  seven  ninths. 


§  160.  It  was  observed,*  that  two  things  are  TWO  things 

.  necessary  to 

necessary  to  the  clear  apprehension  01  an  mte-  apprehends 

,  number. 

ger  number. 

1st.  A  distinct  apprehension  of  the  unit  which  First. 
forms  the  basis  of  the  number  ;  and, 

2dly.  A  distinct  apprehension  of  the  number  second. 
of  times  which  that  unit  is  taken. 

Three  things  are  necessary  to  the  distinct  ap-  Three  things 

.  necessary  to 

prehension  of  the  value  of  any  fraction,  either  apprehend  a 

,       .         ,                 ,  fraction. 

decimal  or  vulgar. 


1st.  We  must  know  the  unit,  or  whole  thing, 
from  which  the  fraction  was  derived  ; 

2d.  We  must  know  into  how  many  equal  parts     second. 
that  unit  is  divided  ;  and, 

3dly.  We  must  know  how  many  such  parts      Third. 
are  taken  in  the  expression. 

The  unit  from  which  the  fraction  is  derived,   unit  of  tho 

fraction  —  of 

is  called  the  unit  of  the  fraction  ;   and  one  of    the  expres- 

the  equal  parts  is  called,  the  unit  of  the  expres 

sion. 

For  example,  to  apprehend  the  value  of  the 

Section  110. 


156  MATHEMATICAL     SCIENCE.  [BOOK  II. 

What  we    fraction  f  of  a  pound  avoirdupois,  or  jib. ;    we 

must  know. 

must  know, 

First  1st.  What  is  meant  by  a  pound  ; 

second.         2d.  That  it  has  been  divided  into  seven  equal 

parts;  and, 
Third.          3d.  That  three  of  those  parts  are  taken. 

In  the  above  fraction,  1  pound  is  the  unit  of 
the  fraction;   one-seventh  of  a  pound,  the  unit 
of  the  expression ;  and  3  denotes  that  three  frac 
tional  units  are  taken. 
Umt  when        If  the  unit  of  a  fraction  be  not  named,  it  is 

not  named.     t    , 

taken  to  be  the  abstract  unit  1. 


ADVANTAGES     OF     FRACTIONAL     UNITS. 

Every  equal       §  161.  By  considering  every  equal  part  of  uni- 

partofone,  a 

unit.  ty  as  a  unit  of  itself,  having  a  certain  relation  to 
the  unit  1,  the  mind  is  led  to  analyze  a  frac 
tion,  and  thus  to  apprehend  its  precise  significa 
tion. 

Advantages  Under  this  searching  analysis,  the  mind  at 
once  seizes  on  the  unit  °f  the  fraction  as  the 
principal  basis.  It  then  looks  at  the  value  of 
each  part.  It  then  inquires  how  many  such  parts 
are  taken. 

Tt  havin§  been  shown  that  eq™l  integer  units 
tegraiorfrac-can  alone  be  added,  it  is  readily  seen  that  the 


CHAP.   II.]  ARITHMETIC ADVANTAGES.  157 

same   principle    is    equally    applicable    to    frac-    tionai,can 
tional    units ;   and   then   the    inquiry  is    made :      added. 
What  is  necessary  in  order  to  make  such  units 
equal  ? 

It  is  seen  at  once,  that  two  things  are  neces-   Two  thin?3 

necessary  for 

sary :  addition. 

1st.  That  they  be  parts  of  the  same  unit ;  and,  Fir9t- 

2d.  That  they  be  like  parts ;  in  other  words,  second. 

they   must   be  of  the    same   denomination,  and 

have  a  common  denominator. 

In  regard  to  Decimal    Fractions,   all   that   is  Decimal 

Fractions. 

necessary,  is  to  observe  that  units  of  the  same 
value  are  added  to  each  other,  and  when  the 
figures  expressing  them  are  written  down,  they 
should  always  be  placed  in  the  same  column. 


§  162.  The  great  difficulty  in  the  management  Difficulty in 

of  fractions,  consists  in  comparing   them  with  ment  of  frac 
tions. 

eack  other,  instead  of  constantly  comparing  them 
with  the  unity  from  which  they  are  derived. 
By  considering  them  as  entire  things,  having  a 

How 

fixed  relation  to  the  unity  which  is  their  basis,  obviated, 
they  can  be  compared  as  readily  as  integer  num 
bers  ;  for,  the  mind  is  never  at  a  loss  when  it 
apprehends  the  unit,  the  parts  into  which  it  is 
divided,  and  the  number  of  parts  which  are 
taken.  The  only  reasons  why  we  apprehend  and 


MATHEMATICAL     SCIENCE.  [BOOK  II. 

handle  integer  numbers  more  readily  than  frac 
tions,  are, 

1st.  Because  the  unity  forming  the  basis  is 
always  kept  in  view ;  and, 

2d.  Because,  in  integer  numbers,  we  have 
been  taught  to  trace  constantly  the  connection 
between  the  unity  and  the  numbers  which  come 
from  it ;  while  in  the  methods  of  treating  frac 
tions,  these  important  considerations  have  been 
neglected. 


SECTION  III. 

PROPORTION     AND     RATIO. 

§  163.  PROPORTION  expresses  the  relation  which 
one  number  bears  to  another,  with  respect  to  its 
being  greater  or  less. 

Two  numbers  may  be  compared,  the  one  wilii 
the  other,  in  two  ways  : 
i«t  method.       1st.  With  respect  to  their   difference,   called 

Arithmetical  Proportion ;  and, 

2d  method.       2d.  With    respect    to    their    quotient,   called 
Geometrical  Proportion. 


CHAP.   II. J  ARITHMETIC PROPORTION.  159 

Thus,  if  we  compare  the  numbers  1  and  8,   Example  of 
by  their  difference,  we  find  that  the  second  ex-  ^^ 
ceeds  the  first  by  7  :  hence,  their  difference  7, 
is  the  measure  of  their  arithmetical  proportion. 

Arithmetical 

and  is  called,  in  the  old  books,  their  arithmetical      Ratio. 
ratio. 

If  we   compare  the  same   numbers   by  their  Exampjeof 

Geometrical 
Proportion. 


quotient,  we  find  that  the  second  contains  the  Geometrical 


first  8  times  :  hence,  8  is  the  measure  of  their 
geometrical  proportion,  and  is  called  their  geo 
metrical  ratio.* 

§  164.  The  two  numbers  which  are  thus  corn- 
Terms. 

pared,  are  called  terms.     The  first  is  called  the  Antecedent. 
antecedent,  and  the  second  the  consequent.  consequent. 

In  comparing  numbers  with  respect  to  their  comparison 
difference,    the    question   is,    how   much   is   one    y 
greater  than  the  other  ?     Their  difference  affords 
the  true  answer,  and  is  the  measure  of  their  pro 
portion. 

In  comparing  numbers  with  respect  to  their  comparison 

.  .       ,  by  quotient. 

quotient,  the  question  is,  how  many  times  is  one 
greater  or  less  than  the  other  ?  Their  quotient 
or  ratio,  is  the  true  answer,  and  is  the  measure 


*  The  term  ratio,  as  now  generally  used,  means  the  quo 
tient  arising  from  dividing  one  number  by  another.  We 
shall  use  it  only  in  this  sense. 


1GO  MATHEMATICAL     SCIENCE.  [fiOOK   II. 


Example  by  of  their   proportion.      Ten,    for   example,   is    9 

difference. 

greater  than  1,  if  we  compare  the  numbers  one 

and  ten  by  their  difference.     But  if  we  compare 

By  quotient,  them  by  their  quotient,  ten   is  said  to  be  ten 

"Ten times."  times  as  great — the  language  "ten  times"  having 

reference  to  the  quotient,  which  is  always  taken 

as  the   measure   of  the   relative   value   of  two 

Examples  of  numbers   so   compared.      Thus,  when   we  say, 

hlst"srem°'the  that,  the  units  of  our  common  system  of  numbers 

increase  in  a  tenfold  ratio,  we  mean  that  they  so 

increase  that  each  succeeding  unit  shall  contain 

the  preceding  one  ten  times.     This  is  a  conven- 

convenient   ient  language  to  express  a  particular  relation  of 

two   numbers,    and  is   perfectly   correct,   when 

used  in  conformity  to  an  accurate  definition. 

in  what         §  165.  All  authors  agree,  that  the  measure  of 

all  authors        ,  . 

agree-  the  geometrical  proportion,  between  two  num 
bers,  is  their  ratio ;  but  they  are  by  no  means 

in  what  disa- unanimous,  nor  does  each  always  agree  with 
himself,  in  the  manner  of  determining  this  ratio. 
Some  determine  it,  by  dividing  the  first  term  by 

Different  me-  the  second ;  others,  by  dividing  the  second  term 

tliods. 

standard  the    y  the  ""*'*     A11  agree' that  the  standard,  what- 
divisor,     ever  it  may  be,  should  be  made  the  divisor. 


*  The  Encyclopedia  Metropolitana,  a  work  distinguished 
by  the  excellence  of  its  scientific  articles,  adopts  the  latter 
method. 


CHAP.   II.]  ARITHMETIC RATIO.  161 

This  leads  us  to  inquire,  whether  the   mind  what  is  the 

best  form. 

fixes  most  readily  on  the  first  or  second  number 
as  a  standard ;  that  is,  whether  its  tendency  is 
to  regard  the  second  number  as  arising  from  the 
first,  or  the  first  as  arising  from  the  second. 

§   166.   All   our   ideas    of   numbers    begin   at     origin  of 

numbers. 

one.*'     This    is    the    starting-point.     We    con 
ceive  of  a  number  only  by  measuring  it  with  How  we  con 
one,  as  a   standard.     One   is   primarily   in   the     ^^ 
mind  before  we   acquire  an  idea   of  any  other 
number.     Hence,   then,   the    comparison   begins   Where  lhe 
at  one,  which  is   the  standard  or  unit,  and  all 
other  numbers  are  measured  by  it.    When,  there 
fore,  we  inquire  what  is  the  relation  of  one  to 
any  other  number,  as  eight,  the  idea  presented    The  idea 
is,  how  many  times  does  eight  contain  the  stand-    Prcsented- 
ard? 

We  measure  by  this  standard,  and  the  ratio  is     standard. 

Ratio. 

the  result  of  the  measurement.     In  this  view  of 

the  case,  the  standard  should  be  the  first  number  ^J^ 

named,  and  the  ratio,  the  quotient  of  the  second 

number  divided  by  the  first.     Thus,  the  ratio  of 

2  to  6  would  be  expressed  by  3,  three  being  the    Example. 

number  of  times  which  6  contains  2. 


*  Section  104. 
11 


102  MATHEMATICAL     SCIENCE.  [BOOK  II. 

other  reasons      §  167.  The  reason  for  adopting  this   method 

for  this  me 

thod  of  com-  of  comparison  will  appear  still  stronger,  if  we 
take  fractional  numbers.  Thus,  if  we  seek  the 
relation  between  one  and  one-half,  the  mind  im 
mediately  looks  to  the  part  which  one-half  is  of 

comparison  onej  ancj  tm's  js  determined  by  dividing  one-half 

of  unity  with 

fractions,  by  1  ;  that  is,  by  dividing  the  second  by  the 
first  :  whereas,  if  we  adopt  the  other  method, 
we  divide  our  standard,  and  find  a  quotient  2. 

Geometrical       §  168.  It  may  be  proper  here  to  observe,  that 

proportion. 

while  the  term  "geometrical  proportion"  is  used 

to  express    the  relation   of  two  numbers,  com- 

A  geometri-  pared  by  their  ratio,  the  term,  "  A  geometrical 

CM!  propor 

tion  defined,  proportion,"  is  applied  to  four  numbers,  in  which 
the  ratio  of  the  first  to  the  second  is  the  same  as 
that  of  the  third  to  the  fourth.  Thus, 

Example.  2  :  4  :  :  6  :   12 

is  a  geometrical  proportion,  of  which  the  ratio 
is  2. 

Further  ad-       §  169.  We  will   now   state   some  further  ad 
vantages. 

vantages  which  result  from  regarding  the  ratio 

as  the  quotient  of  the  second  term  divided  by 
the  first. 


o        Every  <luestion  in  ^   Rule  of   Three  is   a 
Three:      geometrical  proportion,  excepting  only,  that  the 


CHAP.   II.]  ARITHMETIC  -  RATIO.  163 

last  term  is  wanting.     When  that  term  is  found,  Their  nature. 

the  geometrical   proportion   becomes    complete. 

In  all  such  proportions,  the  first  term  is  used  as 

the  divisor.     Further,  for  every  question  in  the 

Rule  of  Three,  we  have  this  clear  and  simple 

solution  :    viz.   that,   the   unknown  term  or  an-  HOW  solved. 

swer,  is  equal  to  the  third  term  multiplied  by 

the  ratio  of  the  first  two.     This  simple  rule,  for 

finding  the  fourth  term,  cannot  be  given,  unless  Thisrule  de- 

pends  on  the 

we  define  ratio  to  be  the  quotient  of  the  second  definition  of 

Ratio. 

term  divided  by  the  first.  Convenience,  there 
fore,  as  well  as  general  analogy,  indicates  this  as 
the  proper  definition  of  the  term  ratio. 


§  170.  Again,   all    authors,   so    far    as  I    have 
consulted  them,  are  uniform  in  their  definition  tl° 


of  the  ratio  of  a  geometrical  progression  :  viz. 
that  it  is  the  quotient  which  arises  from  divid 
ing  the  second  term  by  the  first,  or  any  other 
term  by  the  preceding  one.  For  example,  in 
the  progression 

2  :  4  :  8  :   16  :  32  :  64,  &c.,  Example: 

all  concur  that  the  ratio  is  2  ;  that  is,  that  it  is     in  which 
the  quotient  which  arises  from  dividing  the  sec- 
ond  term  by  the  first  :  or  any  other  term  by  the 
preceding  term.     But  a  geometrical  progression 
diners   from   a   geometrical   proportion  only  in 


164  MATHEMATICAL     SCIENCE.  [fiOOK   II. 


The  same     this  :  in  the  former,  the  ratio  of  any  two  terms 
place  m  every  is  the  same  ',  while  in  the  latter,  the  ratio  of  the 


first  and  second  is  different  from  that  of  the  sec- 
au  the  same.   ond  and  tnird      There  is,  therefore,  no  essential 
difference  in  the  two  proportions. 

Why,  then,  should  we  say  that  in  the  propor 
tion 

2  :  4  :  :  6  :   12, 

the  ratio  is  the  quotient  of  the  first  term  divided 

Examples. 

by  the  second  ;  while  in  the  progression 
2  :  4  :  8  :  16  :  32  :  64,  &c., 

the  ratio  is  defined  to  be  the  quotient  of  the  sec 
ond  term  divided  by  the  first,  or  of  any  term  di 
vided  by  the  preceding  term  ? 
wherein         As  far  as  I  have  examined,  all   the  authors 

authors  ,        ,  .     _ 

have  depart-  wno  nave  denned  the  ratio  of  two  numbers  to 

lefimuolf  be  the  quotient  of  the  first  divided  by  the  sec 

ond,  have  departed  from  that  definition  in   the 

case  of  a  geometrical  progression.     They  have 

HOW  used    there  used  the  word  ratio,  to  express  the  quo 

tient  of  the  second  •  term  divided  by  the  first, 

and  this  without  any  explanation  of  a  change 

in  the  definition. 

other  m-        Most  of  them  have  also  departed  from  their 

stances  in       ,    „    .   .  .        , 

which  the    definition,   m  informing    us    that  "  numbers    in- 

deflnition  of  r 

crease  from  right  to  left  in  a  tenfold  ratio,"  in 


CHAP.  II.]  ARITHMETIC PROPORTION.  165 

which  the  term  ratio  is  used  to  denote  the  quo-   Ratio  is  not 
tient  of  the  second  number  divided  by  the  first. 
The  definition  of  ratio  is  thus  departed  from, 

and   the   idea   of  it   becomes   confused.     Such   consequen 
ces. 
discrepancies   cannot    but    introduce    confusion 

into  the  minds  of  learners.  The.  same  term 
should  always  be  used  in  the  same  sense,  and 
have  but  a  single  signification.  Science  does  what  science 

demands. 

not  permit  the  slightest  departure  from  this  rule. 
I  have,  therefore,  adopted  but  a  single  significa 
tion  of  ratio,  and  have  chosen  that  one  to  which  Thedeflni- 
all  authors,  so  far  as  I  know,  have  given  their 
sanction ;  although  some,  it  is  true,  have  also 
used  it  in  a  different  sense. 


§  171.  One  important  remark  on  the  subject    important 

.  Remark. 

of  proportion  is  yet  to  be  made.     It  is  this  :• 

Any  two  numbers  which  are  compared  togeth-     Numbers 

compared 

er,  either  by  their  difference  or  quotient,  must   must  be  of 

.  .  .  ,  the  same 

be  of  the  same  kind:  that  is,  they  must  either       kind> 
have  the  same  unit,  as  a  basis,  or  be  susceptible 
of  reduction  to  the  same  unit. 

For  example,  we  can  compare  2  pounds  with    Examples 

relating  to 

6  pounds  :  their  difference  is  4  pounds,  and  their  Arithmetical 
ratio  is  the  abstract  number  3.     We  can  also  rical  Propor. 
compare  2  feet  with  8  yards :  for,  .although  the 
unit  1  foot  is  different  from  the  unit  1  yard,  still 
8  yards  are  equal  to  24  feet.     Hence,  the  differ- 


166  MATHEMATICAL     SCIENCE.  [BOOK  II. 

ence  of  the  numbers  is  22  feet,  and  their  ratio 
the  abstract  number  12. 
Numbers         On  the  other  hand,  we  cannot  compare  2  dol- 

•with  different 

units  cannot  lars  with  2  yards  of  cloth,  for  they  are  quantities 
'  of  different  kinds,  not  being  susceptible  of  reduc 
tion  to  a  common  unit. 
Abstract         Simple  or  abstract  numbers  may  always  be 

numbers  may 

be  compared,  compared,  since  they  have  a  common  unit  1 


SECTION  IY. 

APPLICATIONS    OF    THE    SCIENCE    OF    ARITHMETIC. 

§  172.  ARITHMETIC    is    both  a  science  and  an 
Arithmetic:  art.     It  is  a  science  in    all  that  relates  to  the 

In  what  a 

science,     properties,   laws,   and   proportions   of   numbers. 

The  science  is  a  collection  of  those  connected 

science  de-  processes  which  develop  and  make  known   the 

laws  that  regulate  and  govern  all  the  operations 

performed  on  numbers. 

what  the        §  173.  Arithmetic  is  an  art,  in  this  :  the  sci- 

science  per 
forms,      ence  lays  open  the  properties  and  laws  of  num 
bers,  and  furnishes  certain  principles  from  which 


CHAP.  II.]  ARITHMETIC APPLICATIONS.  1G7 

practical  and  useful  rules  are  formed,  applicable 
in  the  mechanic  arts  and  in  business  transac 
tions.  The  art  of  Arithmetic  consists  in  the  in  what  the 

art  consists. 

judicious  and  skilful  application  of  the  princi 
ples  of  the  science ;  and  the  rules  contain  the 
directions  for  such  application. 

§  174.  In  explaining  the  science  of  Arithmetic,  in  explaining 
great  care  should  be  taken  that  the  analysis  of  whatTilc"^a- 
every  question,  and  the  reasoning  by  which  the 
principles  are  proved,  be  made  according  to  the 
strictest  rules  of  mathematical  logic. 

Every  principle  should  be  laid  down  and  ex-    HOW  each 
plained,  not  only  with  reference  to  its  subsequent    ^oukfbe 
use  and  application  in  arithmetic,  but  also,  with      stated> 
reference  to  its  connection  with  the  entire  mathe 
matical  science — of  which,  arithmetic  is  the  ele 
mentary  branch. 

§  175.  That  analysis   of  questions,  therefore,      what 

,        .  ,  .  questions  are 

where  cost  is  compared  with  quantity,  or  quan-  feulty. 
tity  with  cost,  and  which  leads  the  mind  of  the 
learner  to  suppose  that  a  ratio  exists  between 
quantities  that  have  not  a  common  unit,  is,  with 
out  explanation,  certainly  faulty  as  a  process  of 
science. 

For  example  :  if  two  yards  of  cloth  cost  4  dol- 

Example. 

lars,  what  will  6  yards  cost  at  the  same  rate  ? 


1G8  MATHEMATICAL     SCIENCE.  [BOOK   II. 

Analysis:  Analysis. — Two  yards  of  cloth  will  cost  twice 
as  much  as  1  yard  :  therefore,  if  two  yards  of 
cloth  cost  4  dollars,  1  yard  will  cost  2  dollars. 
Again  :  if  1  yard  of  cloth  cost  2  dollars,  6  yards, 
being  six  times  as  much,  will  cost  six  times  two 
dollars,  or  12  dollars. 
satisfactory  Now,  this  analysis  is  perfectly  satisfactory  to 

to  a  child.  , 

a  cnila.  Me  perceives  a  certain  relation  between 
2  yards  and  4  dollars,  and  between  6  yards  and 
12  dollars :  indeed,  in  his  mind,  he  compares 
these  numbers  together,  and  is  perfectly  satisfied 
with  the  result  of  the  comparison. 

Advancing  in  his  mathematical    course,  how 
ever,  he  soon  comes  to  the  subject  of  propor 
tions,    treated    as    a   science.     He    there    finds, 
Reason  why  greatly  to  his  surprise,  that  he  cannot  compare 

it  is  defective. 

together  numbers  which  have  different  units; 
and  that  his  antecedent  and  consequent  must  be 
of  the  same  kind.  He  thus  learns  that  the  whole 
system  of  analysis,  based  on  the  above  method  of 
comparison,  is  not  in  accordance  with  the  prin 
ciples  of  science. 
True  What,  then,  is  the  true  analysis  ?  It  is  this  : 

analysis : 

6  yards  of  cloth  being  3  times  as  great  as  2 
yards,  will  cost  three  times  as  much  :  but  2  yards 
cost  4  dollars  ;  hence,  6  yards  will  cost  3  times 
4,  or  12  dollars.  If  this  last  analysis  be  not 
as  simple  as  the  first,  it  is  certainly  mote  strictly 


tiflc. 


CHAP.   II.]  ARITHMETIC APPLICATIONS.  169 


scientific  ;    and  when  once  learned,  can  be  ap-        its 
plied  through  the  whole  range  of  mathematical 
science. 


§  176.  There  is  yet  another  view  of  this  ques-   Reasons  in 

, .  i  .    |  ,  .  c  favor  of  the 

tion  which  removes,  .to  a  great  degree,  if  not  erst  analysis. 

entirely,  the  objections  to  the  first  analysis.    It  is 

this: 

The  proportion  between  1  yard  of  cloth  and 
its  cost,  two  dollars,  cannot,  it  is  true,  as  the 
units  are  now  expressed,  be  measured  by  a  ratio, 
according  to  the  mathematical  definition  of  a 
ratio.  Still,  however,  between  1  and  2,  regard 
ed  as  abstract  numbers,  there  is  the  same  relation  Numbers 
existing  as  between  the  numbers  6  and  12,  also  mustbere- 

garded  as  ab- 

regarded  as  abstract.     Now,  by  leaving  out  of     stract: 

view,  for   a  moment,  the  units  of  the  numbers, 

and  finding  12  as  an  abstract  number,  and  then  The  analysis 

...  .,  ,  then  correct. 

assigning  to  it  its  proper  unit,  we  have  a  correct 
analysis,  as  well  as  a  correct  result. 


§  177.  It  should  be  borne  in  mind,  that  practi-  How  the 

rules  of  arith- 

cal  arithmetic,  or  arithmetic   as  an  art,   selects  meticare 
from  all  the  principles  of  the  science,  the  mate 
rials  for  the  construction   of  its  rules  and  the 

proofs   of  its  methods.     As   a  mere  branch  of  What 

practical  knowledge,  it  cares  nothing  about  the  ^^  e 

forms  or  methods   of  investigation — it  demands  demands. 


170  MATHEMATICAL    SCIENCE.  LBOOK  H- 


the  fruits  of  them  all,  in  the  most  concentrated 
Best  rule  of  and  practical  form.     Hence,  the  best  rule  of  art, 
^        which  is  the  one  most  easily  applied,  and  which 
reaches  the  result  by  the  shortest  process,  is  not 
always  constructed  after  those   methods  which 
science  employs  in  the  development  of  its  prin 
ciples. 

Definition  of       For  example,  the  definition  of  multiplication  is, 
mition!1Ca    tnat  it  i§  tne  process  of  taking  one  number,  called 
the  multiplicand,  as   many  times    as    there   are 
what  it  de-  units  in  anotner  called  the  multiplier.     This  defi- 
mands.      nition,  as  one  of  science,  requires  two  things. 
First.  1st.  That  the  multiplier  be  an  abstract  num 

ber;  and, 

second.         2dly.  That  the  product  be  of  the  same  kind  as 
the  multiplicand. 

These  two  principles  are  certainly  correct, 
Maybe  and  relating  to  arithmetic  as  a  science,  are  uni- 
versaUy  true-  But  are  they  universally  true,  in 
the  sense  in  which  the7  would  be  understood  by 
learners,  when  applied  to  arithmetic  as  a  mixed 
subject,  that  is,  a  science  and  an  art  ?  Such  an 
application  would  certainly  exclude  a  large  class 
of  practical  rules,  which  are  used  in  the  appli 
cations  of  arithmetic,  without  reference  to  par 
ticular  units. 

Examples  of         For    example>  if  we  haye    ^    in    ^^    ^  ^ 

applications,  multiplied  by  feet  in  height,  we  must  exclude  the 


CHAP.  II.]          ARITHMETIC APPLICATIONS.  171 

question  as  one  to  which  arithmetic  is  not  appli 
cable  ;  or  else  we  must  multiply,  as  indeed  we 
do,  without  reference  to  the  unit,  and  then  assign 
a  proper  unit  to  the  product. 

If  we  have   a  product  arising  from  the  three    w*160  the 

three  factors 

factors    of  length,    breadth,  and    thickness,    the     are  lines, 
unit  of  the  first  product  and  the  unit  of  the  final 
product,   will  not   only  be  different   from  each 
other,  but  both  of  them  will  be  different  from 
the  unit  of  the  given  numbers.     The  unit  of  the  The  different 
given  numbers  will  be  a  unit  of  length,  the  unit 
of  the  first  product  will  be  a  square,  and  that  of 
the  final  product,  a  cube. 

§  178.  Again,  if  we  wish  to  find,  by  the  best      other 

J  examples. 

practical  rule,  the  cost  of  467  feet  of  boards  at 
30  cents  per  foot,  we  should  multiply  467  by 
30,  and  declare  the  cost  to  be  14010  cents,  or 
$140,10. 

Now,  as  a  question  of  science,  if  you  ask,  can   considered 

,   .    ,       (,         ,  0  .    i      as  a  question 

we  multiply  feet  by  cents  f  we  answer,  certainly    ofscience< 
not.     If  you  again    ask,  is   the   result   obtained 
right  ?  we  answer,  yes.    If  you  ask  for  the  analy- 
sys,  we  give  you  the  following : 

1  foot  of  boards  :  467  feet  :  :  30  cents  :  Answer. 

Now,  the  ratio  of  1   foot  to  467  feet,  is  the  ab-      Ratio, 
stract  number  467 ;  and  30  cents  being  multi- 


172  MATHEMATICAL     SCIENCE.  [BOOK  II. 

plied  by  this  number,  gives  for  the  product  14010 
cents.     But  as  the  product  of  two  numbers  is 

Product  of 

two       numerically  the  same,  whichever  number  be  used 

as  the  multiplier,  we  know  that  467  multiplied  by 

30,  gives  the  same  number  of  units  as  30  multi- 

The  first  rule  p\ied  by  467 :  hence,  the  first  rule  for  finding  the 

correct 

amount  is  correct. 


scientific  in-  §  179.  I  have  given  these  illustrations  to  point 
out  the  difference  between  a  process  of  scientific 

Practica     investigation  and  a  practical  rule. 

The  first  should  always  present  the  ideas  of 

Their  difler-  the  subject  in  their  natural  order  and  connection, 

what  it  con-  while  the  other  should  point  out  the  best  way  of 
obtaining  a  desired  result.  In  the  latter,  the 
steps  of  the  process  may  not  conform  to  the  or 
der  necessary  for  the  investigation  of  principles ; 
but  the  correctness  of  the  result  must  be  suscepti 
ble  of  rigorous  proof.  Much  needless  and  un- 

Causesof    Pr°fitable  discussion  has  arisen  on  many  of  the 
error.      processes  of  arithmetic,  from  confounding  a  princi 
ple  of  science  with  a  rule  of  mere  application. 


CHAP.   II.] 


ARITHMETI C ORDER. 


173 


SECTION  Y. 


METHODS    OF    TEACHING    ARITHMETIC    CONSIDERED. 


ORDER    OF    THE    SUBJECTS. 


§>  180.  IT  has  been  well  remarked  by  Cousin, 
the  great  French  philosopher,  that  "  As  is  the 
method  of  a  philosopher,  so  will  be  his  system ; 
and  the  adoption  of  a  method  decides  the  destiny 
of  a  philosophy." 

What  is  said  here  of  philosophy  in  general,  is 
eminently  true  of  the  philosophy  of  mathematical 
science ;  and  there  is  no  branch  of  it  to  which 
the  remark  applies,  with  greater  force,  than  to 
that  of  arithmetic.  It  is  here,  that  the  first  no 
tions  of  mathematical  science  are  acquired.  It 
is  here,  that  the  mind  wakes  up,  as  it  were,  to 
the  consciousness  of  its  reasoning  powers.  Here, 
it  acquires  the  first  knowledge  of  the  abstract — 
separates,  for  the  first  time,  the  pure  ideal  from 
the  actual,  and  begins  to  reflect  and  reason  on 
pure  mental  conceptions.  It  is,  therefore,  of  the 
highest  importance  that  these  first  thoughts  be 
impressed  on  the  mind  in  their  natural  and  proper 


Cousin. 

Method 

decides 

Philosophy. 


True  in 
science. 


Why 

important  in 
Arithmetic. 


First 
thoughts 
should  be 

rightly 
impressed. 


174 


MATHEMATICAL     SCIENCE.  [BOOK   II. 


Faculties  to   order,  so  as  to  strengthen  and  cultivate,  at  the 
be  cultivated.  ^^  ^^  the  faculties  of  apprehension,  discrim 
ination,   and  comparison,  and   also  improve  the 
yet  higher  faculty  of  logical  deduction. 

Firet  point:       §  181.    The   first   point,    then,    in   framing   a 
course  of  arithmetical  instruction,  is  to  deter- 
methodof    mine  the  method  of  presenting  the  subject.     Is 
th^bjed!.   there  any  thing  in  the  nature  of  the  subject  it 
self,  or  the  connection  of  its  parts,  that  points 
out  the  order  in  which  these  parts  should  be 
Laws  of     studied  ?     Do   the   laws   of  science   demand   a 
whLTdo     particular   order ;   or   are   the   parts   so   loosely 

they  require?  connected)    ag    to    render    it    a    matter    of   indiffer- 

ence  where  we  begin  and  where  we  end  ?  A 
review  of  the  analysis  of  the  subject  will  aid  us 
in  this  inquiry. 

Basis  of  the       §  182.  We  have   seen*  that  the  science  of 

science  of  . 

numbers,  numbers  is  based  on  the  unit  1.  Indeed,  the 
in  what  the  whole  science  consists  in  developing,  explain- 

consists.     ing,   and   illustrating    the    laws   by    which,    and 

through  which,  we  operate  on  this  unit.     There 

Three  classes  are  three  classes  of  operations  performed  on  the 

of  operations. 

unit  one. 
e  uie       1st.  To  increase  it  according  to  certain  scales, 


unit. 


*  Section  104. 


CHAP.   II.]        ARITHMETIC  -  INTEGER     UNITS.  175 

forming  the  classes  of  simple  and   denominate 
numbers  ; 

2d.  To  divide  it  in  any  way  we  please,  form-      2d.  TO 

,        ,       .        ,          ,         ,  c          .  divide  it. 

ing  the  decimal  and  vulgar  fractions  ;  and, 

3d.  To  compare  it  with  all  the  numbers  which   3d.  TO  com- 
come  from  it  ;  and  then  those  numbers  with  each 
other.     This  embraces  proportions,  of  which  the 
Rule  of  Three  is  the  principal  branch. 

There  is  yet  a  fourth  branch  of  arithmetic  ;      Fourth 
viz.  the  application  of  the  principles  and  of  the 
rules  drawn  from  them,  in  the  mechanic   arts      Practical 

.  ,.  .  f.  applications; 

and   in   the    ordinary  transactions  of  business. 
This   is   called  the    Art,   or   practical   part,   of     these  the 
Arithmetic.     (See  Arithmetical  Diagram  facing 
page  117.) 

Now,  if  this  analysis  be  correct,  it  establishes     Analysis 

.  ,  .  f,          .   ,  .        establishes 

the  order  in  which  the   subjects  of  arithmetic    the  order. 
should  be  taught. 


INTEGER     UNITS. 

§  183.   We  begin  first  with  the  unit  1,  and  in-     Unitone 
crease  it  according  to  the  scale  of  tens,  forming  JJJJJ*-1^ 
the  common  system  of  integer  numbers.     We   the  scale  of 

tens. 

then  perform  on  these  numbers   the  operations 

of  the  five  ground  rules  ;  viz.  numerate  them,   operations 

add  them,  subtract   them,   multiply   and  divide   * 

them. 


176  MATHEMATICAL     SCIENCE.  [BOOK  II. 

Next  increase      We  next  increase  the  unit  1  according  to  the 

it  according 

to  varying  varying  scales  of  the  denominate  numbers,  and 
thus  produce  the  system,  called  Denominate  or 
Concrete  Numbers ;  after  which  we  perform 
upon  this  class  all  the  operations  of  the  five 
ground  rules. 

What  order       §  184.  It  may  be  well  to  observe  here,  that 

the  law  of        ,       , 

exact  science  tne  ^aw  °*  exact  science  requires  us  to  treat  the 
denominate  numbers  first,  and  the  numbers  of 
the  common  system  afterwards;   for,  the  corn- 
Reason  for    mon  system  is  but  a  variety  of  the  class  of  de- 

this. 

nominate  numbers  ;   viz.  that  variety,  in  which 

the  scale  is   the  scale  of  tens,  and  unvarying. 

Reason  for    But  as  some  knowledge  of  a  subject  must  precede 

departing  . 

from  it.      <M  generalization,  we  are  obliged  to  begin  the 
subject  of  arithmetic  with  the  simplest  element. 


FRACTIONAL     UNITS. 

Divisions  of       §  185.  We  now  pass  to  the  second  class  of 

the  unit. 

operations  on  the  unit  1  ;  viz.  the  divisions  of 
General  me-  it.     Here  we  pursue  the  most  general  method, 
and  divide  it  arbitrarily ;  that  is,  into  any  num 
ber  of  equal  parts.     We  then  observe  that  the 
Method  ao-  division  of  it,  according  to  the  scale  of  tens  is 

cording  to     , 

scale  of  tens.  but  a  particular  case  of  the  general  law  of  di 
vision.      We    then   perform,    on    the    fractional 


CHAP.  II.]  ARITHMETIC RATIO.  177 


units  which  thus  arise,  all  the  operations  of  the   operations 

performed. 

rive  ground  rules. 


RATIO, OR     RULE     OF     THREE. 

§  186.  Having  considered  the  two  subjects  of     subjects 

considered. 

integer  and  fractional  units,  we  come  next  to 
the  comparison  of  numbers  with  each  other. 

This  branch   of  arithmetic   develops   all   the    what  this 

.  .  .  branch  do- 

relative   properties   of  numbers,  resulting   from      velops. 
their  inequality. 

The  method  of  arrangement,  indicated  above,  what  the  ar 

.  rangement 

presents  all  the  operations  of  arithmetic  in  con-       does. 
nection  with  the  unit  1,  which  certainly  forms 
the  basis  of  the  arithmetical  science. 

Besides,  this  arrangement  draws  a  broad  line  what  it  does 
between  the  science  of  arithmetic  and  its  ap 
plications  ;   a  distinction  which  it   is  very  im 
portant  to  make.     The  separation  of  the  prin-  Theory  and 

.    ,  T          •  practice 

ciples  of  a   science  from  their  applications,  so    8houidbe 
that  the  learner  shall  clearly  perceive  what  is    sepan 
theory  and  what  practice,  is  of  the  highest  im 
portance.     Teaching  things  separately,  teaching  Golden  rules 

,  .  .for  teaching. 

them  well,  and  pointing  out  their  connections, 
are  the  golden  rules  of  all  successful  instruc 
tion. 


§187.  I  had  supposed,  that  the  place  of  the 
12 


178  MATHEMATICAL     SCIENCE.  [BOOK  II. 

Rule  of  Three,  among  the  branches  of  arith 
metic,  had  been  fixed  long  since.  But  several 

Differences  in  authors  of  late,  have  placed  most  of  the  practi- 

'  cal  subjects  before  this  rule — giving  precedence, 

for  example,  to  the  subjects  of  Percentage,  In- 

in  what  they  terest,  Discount,  Insurance,  &c.     It  is  not  easy 

consist.  . 

to  discover  the  motive   ot    this  change.     It  is 
Ratio  part  of  certain  that  the  proportion  and  ratio  of  num- 

the  science. 

bers  are  parts  of  the  science  of  arithmetic ;  and 
should  pre-   the  properties  of  numbers   which   they  unfold, 

cede  applica 
tions,       are  indispensably  necessary  to  a  clear  apprehen 
sion  of  the  principles  from  which  the  practical 
rules  are  constructed. 

We  may,  it  is  true,  explain  each  example  in 

Percentage,  Interest,  Discount,  Insurance,  &c., 

cannot  wen   by  a  separate  analysis.     But    this    is  a  matter 

ch  tinge  th6 

order.      of  much  labor ;   and  besides,  does  not  conduct 

the   mind   to   any  general   principle,   on   which 

all  the  operations  depend.     Whereas,  if  the  Rule 

of  Three  be  explained,  before  entering  on  the 

Advantages   practical  subjects,  it  is  a  great  aid  and  a  pow- 

of  first  ex-  c  ,  .,. 

plaining  the  crtul    auxiliary   in    explaining    and    establishing 

TnL0.f     a11  the  Poetical  rules.     If  the   Rule  of  Three 

is    to   be    learned    at    all,    should    it   not   rather 

precede  than  follow  its   applications?      It  is  a 

great  point,  in  instruction,  to  lay  down  a  gen- 

J^ipTof   eral  PrinciPle>  as  early  as  possible,  and  then  con- 

induction,   nect  with  it,  and  with  each  other,  all  the  subor- 


CHAP.  II.]       ARITHMETIC  -  PRACTICAL     PART.  179 

dinate  principles,  with  their  applications,  which 
flow  from  it. 

PRACTICAL    PART. 


§  188.  We   come  next   to   the  4th  division; 

of  arithmetic. 

viz.  the  applications  of  arithmetic. 

Under  the  classification  which  we  have  indi-    Whathaa 

been  done. 

cated,  all  the  principles  of  the  science  will  have 
been  mastered,  when  the  pupil  reaches  this  stage 
of  his  progress.  His  business  will  now  be  with  What 

remains  to  be 

the  application  of  principles,  and  no  longer  in      done. 
the    study   and    development   of  the   principles 
themselves.     The  unity  and   simplicity  of  this  ^^^ 
method  of  classification,  may  be  made  more  ap 
parent,  by  the  aid  of  the  arithmetical  diagram 
which  faces  page  117. 

May  we  not  then  conclude  that  the  subjects  H°wthesub- 

*  jects  should 

of  arithmetic  should  be  presented  in  the  follow-  be  presented. 
ing  order  : 

1st.  All  the  methods  of  treating  integer  num-  lst-  Integer 

numbers. 

bers,  whether  formed  from  the  unit  1  according 
to  the  scale  of  tens,  or  according  to  varying 
scales  ; 

2d.  All  the  methods  of  treating  fractional  uni-    2d.  Frac 

tions. 
ties,  whether  derived  from  the  unit  1  according 

to  the  scale  of  tens,  or  according  to  varying 
scales  ; 


180       '  MATHEMATICAL     SCIENCE.  [BOOK  II. 


3d.  Rule  of       3d.   The  proportion  and  ratios   of  numbers; 

Three. 

and, 

4th.APPiica-  4th.  The  applications  of  the  science  of  num 
bers  to  practical  and  useful  objects. 

OBJECTIONS    TO    THIS    CLASSIFICATION     ANSWERED. 

§  189-  I*  nas  been  urged  that  Common  or  Vul- 
Sar  Fractions  should  be  placed  "immediately 
after  Division,  for  two  reasons." 

"  First,  they  arise  from  division,  being  in  fact 
unexecuted  division." 

"Second,    in   Reduction    and   the    Compound 

Second. 

Rules,  it  is  often  necessary  to  multiply  and  divide 
fractions,  to  add  and  subtract  them,  also  to  carry 
for  them,  unless  perchance  the  examples  are  con 
structed  for  the  occasion,  and  with  special  refer 
ence  to  avoiding  these  difficulties." 

These  are  aii.  These,  I  believe,  are  all  the  objections  that 
have  been,  or  can  be  urged  against  the  classifi 
cation  which  I  have  suggested.  I  give  them  in 

Given  in  fuii.  full,  because  I  wish  the  subject  of  arrangement 

to  be  fully  considered  and  discussed.     It  should 

what      be  our  main  object  to  get  at  the  best  possible 

should  be  /•!•/•• 

our  object.    system  ol   classification,   and  not  to  waste   our 

efforts  in  ingenious  arguments  in  the  support  of 

TO  be  con-    a  favorite  one.     We  will  consider  these  obiec- 

aidered  se 
parately,      tions  separately. 


CHAP.  II.]  ARITHMETIC OBJECTIONS.  181 


It  is  certainly  true,  that  fractions  "  arise  from     Fractions 

arise  from  di- 

division,    but  it  is  as  certainly  not  true,  that  they      vision. 
are  "  unexecuted  divisions  ;"    and  this  last  idea 
has  involved  the  subject  in  much  perplexity  and 
difficulty. 

The  most  elementary  idea  of  a  fraction,  arises  The  element 

ary  idea  is 

from  the  division  of  a  single  thing  into  two  equal  obtained  by 
.parts,  each  of  which  is  called  a  half.     Now,  we 


get  no  idea  of  this  half  unless  we  consider  the 
division  perfected.  And  indeed,  the  method  of 
teaching  shows  this.  For,  we  cannot  impress 
the  idea  of  a  half  on  the  mind  of  a  child,  until  Example; 
we  have  actually  divided  in  his  presence  the 
apple  (or  something  else  regarded  as  a  unit), 
and  exhibited  the  parts  separately  to  his  senses  ; 
and  all  other  fractions  must  be  learned  by  a  like 
reference  to  the  unit  1.  Hence,  we  can  form  no  And  not 

.   .  otherwise. 

notion  of  a  fraction,  except  on  the  supposition  ot 
a  perfected  division. 

If  the  term,  "unexecuted  division,"  applies  to  "Unexecuted 

division"does 

the  numerator  of  the  expression,  and  not  to  the  not  apply  to 

,.      ,         ,,          .  i        .  -,          •..•II  •         the  numera- 

unit  of  the  fraction,   the  idea  is  still  more  m-        tor 
volved.     For,  nothing  is  plainer   than   that  we 
can  form  no  distinct  notion  of  a  result,  so  long 
as  the  process  on  which  it  depends  cannot   be 
executed.     The  vague  impression  that  there  is 

That  a  frao- 

something  hanging  about  a  fraction  that  cannot   tion  cannot 
be  quite  reached,  has  involved  the  subject  in  a  rcacAed,has 


182  MATHEMATICAL     SCIENCE.  [BOOK  II. 

occasioned    mysterious   terror ;  and  the   boy   approaches   it 
sulty'     with  the  same  feeling  which  a  mariner  does  a 
rocky  and    dangerous  coast,    of  which   he   has 
neither  map  nor  chart  to  guide  him.     But  pre 
sent  to  the  mind  of  the  pupil  the  distinct  idea, 
that  a  fraction  is  one  or  more    equal  parts    of 
unity,  and  that  every  such  part  is  a  perfect  whole, 
axed  relation  ^avinrr  a  certain  relation  to  the  thing  from  which 

to  unity. 

it  was  derived,  and  all  the  mist  is  cleared  away, 
and  his  mind  divides  the  unit  into  any  number 
of  equal  parts,  with  the  same  facility  as  the  knife 
divides  the  apple. 

Form  the         The  form  of  expression  for  a  fraction,  and  for 

wTunexecu-  an  unexecuted  division,  is  indeed  the  same,  but 

ted  division.  ^  interpretation  of  this  expression,  as  used  for 

one  or  the  other,  is  entirely  different.     In  our 

A  sign  may   common   language,   the   same   word   is   not   al- 

express  dif 
ferent  things,  ways  the  sign  of  the  same  idea ;  and  in  science, 

the  same  symbol  often  expresses  very  different 
things, 
Example         For  example,  |^,  as  an  expression  in  fractions, 

illustrating 

heseprinci-  means,  that  something  regarded  as  a  wnoie  has 
been  divided  in  7  equal  parts,  and  that  3  of  those 
parts  are  taken.  As  a  result  of  division,  it  means 
that  the  integer  number  3  is  to  be  divided  into 

what  cannot  7  equal  parts.     Now,  it  cannot  be  assumed,  as  a 

be  assumed. 

self-evident  fact,  that  three  of  the  parts  of  the 
first  division  are  equal  to  1  part  of  the  second ; 


CHAP.   II.]  ARITHMETIC  -  OBJECTIONS. 


183 


and  if  this  fact  be  made  the  basis  of  a  system 

of  fractions,  the  mind  of  a  child  will  go  through  The^isof 

every  system 

that  system  in  the  dark.     The  basis  of  every  sys-  should  be  HH 

elementary 

tern  should  be  an  elementary  idea.  idea. 


§  190.  The  second  objection,  as  far  as  it  goes, 
is  valid.  In  all  the  tables  of  denominate  num 
bers,  fractions  occur  five  times  ;  viz.  twice  in 
Long  Measure,  where  5^  yards  make  1  rod,  and 
69J  statute  miles  1  degree  ;  once  in  Cloth  Mea 
sure,  where  2-j-  inches  make  1  nail;  once  in 
Square  Measure,  where  30J  square  yards  make 
1  square  rod  ;  and  once  in  Wine  Measure,  where 
31^  gallons  make  1  barrel.  Now,  it  were  a  little 
better,  if  these  tables  had  been  constructed  with 
integer  units.  But  it  should  be  borne  in  mind, 
that  the  first  notions  of  fractions  are  given  either 
by  oral  instruction,  or  learned  from  elementary 
arithmetics.  Most  of  the  leading  arithmetics 
are,  I  believe,  preceded  by  smaller  works.  These 
are  designed  to  impart  elementary  ideas  of  num- 
bers,  so  as  not  to  embarrass  the  classification  of 
subjects  when  the  scholar  is  able  to  enter  on  a 
system.  Now,  the  most  elementary  of  these 
works  conducts  the  pupil,  in  fractions,  far  be 
yond  the  point  necessary  to  understand  and 
manage  all  the  fractions  which  appear  in  the 
tables  of  denominate  numbers;  and  hence,  there 


second  objec- 

tion  valid  * 


But  of  no 


Reasons. 


Design  of 
wortoi 


taught  in  the 

elementary 

works; 


I 

184  MATHEMATICAL     SCIENCE.  [l3OOK    II. 


May  then  be  is  no  reason,  on  that  account,  to  depart  from  a 
classification  otherwise  desirable. 


OBJECTIONS     TO     THE     NEW     METHOD. 

§  191.  Having  examined  the  objections  that 
have  been  urged  against  that  system  of  classifi 
cation  of  the  subjects  of  arithmetic,  which  has 

Objections  to 

the  new  me-  appeared  to  me  most  in  accordance  with    the 

thod  consid-  . 

ered.       principles  of  science,  I  shall  now  point  out  some 
of  the  difficulties  to  be  met  with  in  the  adoption 
of  the  method  proposed  as  a  substitute. 
First  objec-       1st.  That  method  separates  the  simple  and  de- 

tiou 

nominate  numbers,  which,  in  their  general  form 
ation,  differ  from  each  other  only  in  the  scale 
by  which  we  pass  from  one  unit  of  value  to  an 
other, 
second  objeo-      2d.  By  thus  separating  these  numbers,  it  be- 

tion. 

comes  more  difficult  to  point  out  their  connec 
tion  and  teach  the  important  fact,  that  in  all 
their  general  properties,  and  in  all  the  opera 
tions  to  be  performed  upon  them,  they  differ 
from  each  other  in  no  important  particular. 
Third  objec-  3d.  By  placing  the  denominate  numbers  after 

tion ;          T_ 

Vulgar  Fractions,  all  the  principles  and  rules  in 
limitation  of  Fractions  are  limited  in  their  application  to  a 

the  rules. 

single  class  of  fractions  ;  viz.  to  those  fractions 
which  have  the  same  unit 


CHAP.   II.]  ARITHMETIC OBJECTIONS.  185 


For   example,  the  common  rule  for   addition   Examples; 
of  fractions,  under  this  classification,  is,  in  sub 
stance,  the  following  :  "  Reduce  the  fractions  to 
a  common  denominator;   add  their  numerators,    Rule;  not 
and  place    the   sum    over  the   common    denomi 
nator.'' 

As   the  subject   of  denominate   numbers  has  Have  not  yet 

.  ii       •  considered 

not  yet  been  reached,  no  allusion  can  be  made    fractions 

to  fractions  having  different  units.     If  the  learn- 

er  should  happen  to  understand  the  rule  literally, 

he  would  conclude  that,  the  sum  of  all  fractions 

having  a  common  denominator  is  found  by  sim 

ply  adding   their   numerators   and    placing   the    The  rules 

therefore  ap- 

sum  over  the  common  denominator.     But  this    Piytoone 

c  ,  .  c          n  -,     c    class  of  frac- 

cannot,  of  course,  be  so,  since  £  of  a  £  and  f 


of  a  shilling  make  neither  one  pound  nor  one 
shilling. 

What  appears  to  me   most   objectionable   in  Greatest  ob- 

•  M  i        •  jection. 

this  method,  is  this  :  it  fails  to  present  the  im 
portant  fact,  that  no  two  fractions  can  be  blend 
ed  into  one,  either  by  addition  or  subtraction, 
unless  they  are  parts  of  the  same  unit,  as  we7 
as  like  parts. 

By  this  method  of  classification  most  of  the  This  method 

~  r  ofclassifica- 

difncult  questions  which   arise  in  fractions  are   tion  avoida 
avoided,  or  else  the  subject  must   be  resumed   thedifflcult 

J  questions. 

after   denominate   numbers,    and   that   class   of 
questions  treated  separately. 


186  MATHEMATICAL     SCIENCE.  [fiOOK   II. 

What  they        The  class  of  questions  to  which  I  refer,  em 
braces  examples  like  the  following : 

Add  f  of  a  day,  TV  of  an  hour,  and  f  of  a  sec 
ond  together. 

It  is  certainly  true  that  a  boy  will  make  mar 
vellous  progress  in  the  text-book,  if  you  limit 

The  subject  him  to  those  examples  in  which   the  fractions 

posed  of,  but  have  a  common  unit.  But,  will  he  ever  un- 
irnt<  derstand  the  science  of  fractions  unless  his  mind 
be  steadily  and  always  turned  to  the  unit  of  the 
fraction,  as  the  basis  ?  Will  he  understand  the 
value  of  one  equal  part,  so  as  to  compare  and 
unite  it  with  another  equal  part,  unless  he  first 
apprehends,  clearly,  the  units  from  which  those 
parts  were  derived  ? 

Laetobjec-       4th.  By  placing  the  Denominate  Numbers  be- 

tioD  stated. 

tween  Vulgar  and  Decimal  Fractions,  the  gen 
eral  subject  of  fractional   arithmetic  is  broken 
into  fragments.     This  arrangement  makes  it  dif- 
Difficuityof  ficult  to  realize  that  these  two  systems  of  num- 
connection  of  bers  differ  from  each  other  in  no  essential  par 
ticular  ;  that  they  are  both  formed  from  the  unit 
one  by  the  same  process,  with  only  a  slight  mod 
ification  of  the  scale  of  division. 


CHAP.  II.]  ARITHMETIC LANGUAGE.  187 


ARITHMETICAL     LANGUAGE. 

§  192.  We  have  seen  that  the  arithmetical  al-  Arithmetical 

alphabet. 

phabet  contains  ten  characters.  From  these 
elements  the  entire  language  is  formed ;  and  we 
now  propose  to  show  in  how  simple  a  manner. 

The  names  of  the  ten  characters  are  the  first  Names  of  the 

characters. 

ten  words  of  the  language.     If  the   unit  1   be 

added  to  each  of  the  numbers  from  1  to  10  in-     First  ten 

combina- 

clusive,  we  find  the  first  ten   combinations  in       tions. 

arithmetic.!     If  2   be    added,   in   like   manner, 

we  have  the  second  ten  combinations ;   adding  second  ten, 

and  so  on  for 

3,  gives  us  the  third  ten  combinations;  and  so      others, 
on,  until  we  have  reached  one   hundred  com 
binations  (page  123). 

Now,  as  we  progressed,  each  set  of  combina-  Each  set  giv 
ing  one  addi 
tions   introduced  one   additional  word,   and   the  tionaiword. 

results  of  all  the  combinations  are  expressed  by 
the  words  from  two  to  twenty  inclusive. 


§  193.   These  one  hundred  elementary  com-  AII  that  need 

be  commit- 

binations,   are   all   that   need   be   committed    to    ted  tome- 
memory  ;  for,  every  other  is  deduced  from  them. 
They  are,  in  fact,  but  different  spellings  of  the 
first  nineteen  words  which  follow  one.    If  we  ex 
tend  the  words  to  one  hundred,  and  recollect  that 


*  Section  114.  t  Section  116. 


188  MATHEMATICAL     SCIENCE.  [BOOK  II. 


at  one  hundred,  we  begin  to  repeat  the  numbers, 

words  to  be  we  see  that  we  have  but  one  hundred  words  to 

for  addition,  be  remembered  for  addition;   and  of  these,  all 

only  ten     above    ten    are    derivative.      To    this    number, 

uve.       must  of  course  be  added  the  few  words  which 

express  the  sums  of  the  hundreds,  thousands,  &c. 

subtraction:  §  194.  In  Subtraction,  we  also  find  one  hun 
dred  elementary  combinations;  the  results  of 
which  are  to  be  read.*  These  results,  and  all 

Number  of  the  numbers  employed  in   obtaining  them,   are 

words. 

expressed  by  twenty  words. 
Muitipiica-        §  195.  In  Multiplication  (the  table  being  car- 

tion  i 

ried  to  twelve),  we  have  one  hundred  and  forty- 

four   elementary   combinations,!    and   fifty-nine 

Number  of   separate  words  (already  known)  to  express  the 

words. 

results  of  these  combinations. 

Division:         §  196.  In  Division,  also,  we  have  one  hundred 
Number  of   an(*   f°rty-f°ur    elementary   combinations,  f    but 
words.      use  on]v  twe}ve  Words  to  express  their  results. 

Four  hun- 

§197  '•  Thus,  we  have  four  hundred  and  eigh- 
tv-ei£ht  elementary  combinations.  The  results 
of  these  combinations  are  expressed  by  one  hun- 
^re(^  words  ;  viz.  nineteen  in  addition,  ten  in  sub- 
traction»  fifty-nine  in  multiplication,  and  twelve 


tion, 
59  in  multi- 


plication,         *  Section  120.         f  Section  122.         \  Section  123. 


CHAP.  II.]  ARITHMETIC LANGUAGE.  189 

in  division.  Of  the  nineteen  words  which  are  12  in  division, 
employed  to  express  the  results  of  the  combina 
tions  in  addition,  eight  are  again  used  to  express 
similar  results  in  subtraction.  Of  the  fifty-nine 
which  express  the  results  of  the  combinations 
in  multiplication,  sixteen  had  been  used  to  ex 
press  similar  results  in  addition,  and  one  in 
subtraction ;  and  the  entire  twelve,  which  ex 
press  the  results  of  the  combinations  in  division, 
had  been  used  to  express  results  of  previous 
combinations.  Hence,  the  results  of  all  the  ele 
mentary  combinations,  in  the  four  ground  rules, 
are  expressed  by  sixty- three  different  words ;  and  sixty-three 

different 

they  are  the  only  words  employed  to  translate  words  in  uii. 
these  results  from  the  arithmetical  into  our  com 
mon  language. 

The  language  for  fractional  units  is  similar    Language 

_  the  same  for 

in  every  particular.     By  means  of  a  language     fracuOU8. 
thus  formed  we  deduce  every  principle  in  the 
science  of  numbers. 

§  198.  Expressing  these  ideas  and  their  com 
binations  by  figures,  gives  rise  to  the  language  Language  of 

arithmetic: 

of  arithmetic.     By  the  aid  of  this  language  we 

not  only  unfold   the  principles  of  the  science,  its  value  and 

but    are   enabled    to    apply   these    principles  to 

every  question  of  a  practical  nature,  involving 

the  use  of  figures. 


190  MATHEMATICAL     SCIENCE.  [fiOOK  II. 


But  few         §  199.   There  is  but  one  further  idea  to  be 

combinations  .        .       .         .  .  , 

which  presented  :  it  is  this, — that  there  are  very  few 
sT'nufcation  combinations  made  among  the  figures,  which 
or  the  figures,  change,  at  all,  their  signification. 

Selecting  any  two  of  the  figures,  as  3  and  5, 
Examples.    ^O1*  example,  we  see  at  once  that  there  are  but 
three   ways   of  writing   them,   that   will   at   all 
change  their  signification. 
First:          First,  write  them  by  the  side  of  each  )    3  5, 

other )    5  3. 

second:         Second,   write   them,  the   one   over  i      f, 
the  other J      f . 

Third.          Third,  place  a  decimal  point  before  )      .3, 
each )       5 

Now,  each  manner  of  writing  gives  a  differ 
ent  signification  to  both  the  figures.     Use,  how- 
Learn  the    ever,  has  established  that  signification,  and  we 

language  by    T 

^        Know  it,  as  soon  as  we  have  learned  the  lan 
guage. 

We  have  thus  explained  what  we  mean  by 
the    arithmetical    language.     Its    grammar    em 
its  grammar:  braces    the   names  of  its   elementary  signs,   or 
Alphabet-   Alphabet,  —  the   formation    and   number    of  its 

words,  and 

their  uses,  words, — and  the  laws  by  which  figures  are  con 
nected  for  the  purpose  of  expressing  ideas.  We 
feel  that  there  is  simplicity  and  beauty  in  this 
system,  and  hope  it  may  be  useful. 


CHAP.   II.]  ARITHMETIC DEFINITIONS.  191 


NECESSITY    OF    EXACT-  DEFINITIONS    AND    TERMS. 

§  200.   The  principles   of  every  science   are  Principles  of 
a  collection  of  mental  processes,  having  estab 
lished  connections  with  each  other.     In  every 
branch    of    mathematics,    the    Definitions    and   Definitions 

and  terms  : 

Terms  give  form  to,  and  are  the  signs  of,  cer 
tain  elementary  ideas,  which  are  the  basis  of 
the  science.  Between  any  term  and  the  idea 
which  it  is  employed  to  express,  the  connection 
should  be  so  intimate,  that  the  one  will  always 
suggest  the  other. 

These  definitions  and  terms,  when  their  sig-  when  once 

fixed  must 

nifications  are  once  fixed,  must  always  be  used    always  be 


in  the  same  sense.  The  necessity  of  this  is  most 
urgent.  For,  "in  the  whole  range  of  arithmetical 
science  there  is  no  logical  test  of  truth,  but  in  Reason. 
a  conformity  of  the  reasoning  to  the  definitions 
and  terms,  or  to  such  principles  as  have  been 
established  from  them." 

§  201.   With  these  principles,  as   guides,  we    Definitions 

...     .    .  and  terms 

propose  to  examine  some  01  the  definitions  and  examined. 
terms  which  have,  heretofore,  formed  the  basis 
of  the  arithmetical  science.  We  shall  not  con 
fine  our  quotations  to  a  single  author,  and  shall 
make  only  those  which  fairly  exhibit  the  gen 
eral  use  of  the  terms. 


192  MATHEMATICAL    SCIENCE.  [BOOK   II. 

It  is  said, 
Number de-       "Number  signifies  a  unit,  or  a  collection  of 

fined.  .,     ,, 

units. 

HOW  "  The  common  method  of  expressing  numbers 

expressed.  .^  ^  ^  Arabic  Notation.  The  Arabic  method 
employs  the  following  ten  characters,  or  figures," 
&LC. 

Names  of  the  "The  first  nine  are  called  significant  figures, 
because  each  one  always  has  a  value,  or  denotes 
some  number." 

And  a  little  further  on  we  have, 
Figures  have      "  The  different  values  which  figures  have,  are 

values. 

called  simple  and  local  values." 

The  definition  of  Number  is  clear  and  cor- 
Number     rect.     It  is  a  general  term,  comprehending  all 

rightly  de-        ,  ,  .  . 

fined:       the  phrases  which  are  used,   to  express,  either 
separately  or  in  connection,  one  or  more  things 
Also  figures,  of  the  same  kind.     So,  likewise,  the  definition 
of  figures,  that  they  are  characters,  is  also  right. 

Definition  de-      But  mark  how  soon  these  definitions  are  de 
parted  from.  ,   ,,  ._ . 

parted  from.  The  reason  given  why  nine  of  the 
figures  are  called  significant  is,  because  "  each 
one  always  has  a  value,  or  denotes  some  num 
ber."  This  brings  us  directly  to  the  question, 
Has  a  figure  whether  a  figure  has  a  value;  or,  whether  it  is 

value '( 

a  mere  representative  of  value.     Is  it  a  number 
or  a  character  to  represent   number?     Is   it   a 

It  is  merely 

a  character:  quantity  or  symbol?     It  is  defined  to  be  a  char- 


CFIAP.   II.]  ARITHMETIC DEFINITIONS.  193 

acter  which  stands  for,  or  expresses  a  number. 
Has  it  any  other  signification  ?  How  then  can 
we  say  that  it  has  a  value — and  how  is  it  possi-  Has  no  value 
ble  that  it  can  have  a  simple  and  a  local  value  ? 
The  things  which  the  figures  stand  for,  may 
change  their  value,  but  not  the  figures  them 
selves.  Indeed,  it  is  very  difficult  for  John  to 
perceive  how  the  figure  2,  standing  in  the  sec-  but  stands 

-  for  value. 

ond  place,  is  ten  times  as  great  as  the  same  fig 
ure  2  standing  in  the  first  place  on  the  right! 
although  he  will  readily  understand,  when  the 
arithmetical  language  is  explained  to  him,  that 
the  UNIT  of  one  of  these  places  is  ten  times  as  unit  of  place, 
great  as  that  of  the  other. 

§  202.  Let  us  now  examine  the  leading  defi-  Leading  defi 
nition  or  principle  which  forms  the  basis  of  the 
arithmetical  language.     It  is  in  these  words : 

"  Numbers  increase  from  right  to  left   in   a  of  number. 
tenfold  ratio  ;  that  is,  each  removal  of  a  figure 
one  place   towards   the  left,  increases   its   value 
ten  times." 

Now,  it  must  be  remembered,   that   number     Does  not 

,  T  ,    ~        ,  .       .r   .  .  agree  with 

has   been   defined   as   signifying      a  unit,  or  a    the  deflni. 
collection  of  units."     How,  then,  can  it  have  a   t 
right  hand,  or  a  left  ?  and  how  can  it  increase 
from  right   to    left   in  a  tenfold  ratio?"     The 
explanation   given   is,   that   (ieach  removal  of  a 

13 


194  MATHEMATICAL     SCIENCE.  [BOOK  II. 

Explanation,  figure  one  place  towards  the  left,  increases  its 
value  ten  times" 

Number,  signifying  a  collection  of  units,  must 
increase  of   necessarily  increase   according   to   the   law  by 

numbers  has  .  . 

which  these  units  are  combined ;  and  that  law 


ire3'  of  increase,  whatever  it  may  be,  has  not  the 
slightest  connection  with  the  figures  which  are 
used  to  express  the  numbers. 

Ratio.          Besides,   is    the   term   ratio    (yet   undefined), 
one  which  expresses  an  elementary  idea  ?     And 
"Tenfold    is  the  term,  a  "  tenfold  ratio,"  one  of  sufficient 
simplicity  for  the  basis  of  a  system  ? 

Does,  then,  this  definition,  which  in  substance 
is  used  by  most  authors,  involve  and  carry  to 
Four  leading  the  mind  of  the  young  learner,  the  four  leading 
ideas  which  form  the  basis  of  the  arithmetical 


notation  ?  viz.  : 
First.  1st.  That  numbers  are  expressions  for  one  or 

more  things  of  the  same  kind. 
second.         2d.  That  numbers   are  expressed  by  certain 

characters    called  figures ;   and  of  which   there 

are  ten. 
Third.          3d.    That   each   figure    always    expresses    as 

many  units  as  its  name  imports,  and  no  more. 
Fourth.          4th.  That  the  kind  of  thing  which   a  figure 

expresses  depends  on  the  place  which  the  figure 

occupies,  or  on  the  value  of  the  units,  indicated 

in  some  other  way. 


CHAP.   II.]  ARITHMETIC DEFINITIONS. 


195 


Addition: 


First. 


PLACE  is  merely  one  of  the  forms  of  language      Place; 
by  which  we  designate  the  unit  of  a  number,    its  office, 
expressed  by  a  figure.     The  definition  attributes 
this  property  of  place  both  to  number  and  fig 
ures,  while  it  belongs  to  neither. 

§  203.  Having  considered  the  definitions  and 
terms  in  the  first  division  of  Arithmetic,  viz.  in 
notation  and  numeration,  we  will  now  pass  to  Definitions  in 
the  second,  viz.  Addition. 

The  following  are  the  definitions  of  Addition, 
taken  from  three  standard  works  before  me : 

"  The  putting  together  of  two  or  more  num 
bers  (as  in  the  foregoing  examples),  so  as  to 
make  one  whole  number,  is  called  Addition,  and 
the  whole  number  is  called  the  sum,  or  amount." 

"ADDITION  is  the  collecting  of  numbers   to-     second, 
gether  to  find  their  sum." 

"  The  process  of  uniting  two  or  more  num-      Third. 
bers  together,  so  as  to  form  one  single  number, 
is  called  ADDITION." 

"  The  answer,  or  the  number  thus  found,  is 
called  the  sum,  or  amount." 

Now,  is  there  in  either  of  these   definitions     Defects, 
any   test,  or   means   of  determining  when   the 
pupil  gets  the  thing  he  seeks  for,  viz.  "  the  sum 
of  two  or  more  numbers  ?"     No  previous  defi 
nition  has  been  given,  in   either  work,  of  the 


196  MATHEMATICAL     SCIENCE.  [BOOK  II. 

term  SUM.     How  is  the  learner  to  know  what 
he  is  seeking  for,  unless  that  thing  be  defined  ? 
NO  prin-         Suppose  that  John  be  required  to  find  the  sum 
BbudanL    of  the  numbers  3  and  5,  and  pronounces  it  to 
be  10.     How  will  you  correct  him,  by  showing 
that  he  has  not  conformed  to  the  definitions  and 
rules  ?     You  certainly  cannot,  because  you  have 
established  no  test  of  a  correct  process. 

But,  if  you  have  previously  defined  SUM  to  be 
a  number  which  contains  as  many  units  as  there 
are  in  all  the  numbers  added :  or,  if  you  say, 
correct defl-       "Addition  is  the  process  of  uniting  two  or 

nition ; 

more  numbers,  in  such,  a  way,  that  all  the  units 
which  they  contain  may  be  expressed  by  a  sin 
gle  number,  called  the  sum,  or  sum  total ;"  you 
will  then  have  a  test  for  the  correctness  of  the 
Gives  a  test,  process  of  Addition ;  viz.  Does  the  number, 
which  you  call  the  sum,  contain  as  many  units 
as  there  are  in  all  the  numbers  added  ?  The 
answer  to  this  question  will  show  that  John  is 
wrong. 


Definitions  of      §  204.   I  will   now   quote    the   definitions   of 

fractions.       -p,  . 

Tractions   from   the   same  authors,   and  in   the 
same  order  of  reference. 

First  "  We  have  seen,  that  numbers  expressing  whole 
things,  are  called  integers,  or  whole  numbers  ; 
but  that,  in  division,  it  is  often  necessary  to 


CHAP.  II.]  ARITHMETIC  -  DEFINITIONS.  197 

divide  or  break  a  whole  thing  into  parts,  and 
that  these  parts  are  called  fractions,  or  broken 
numbers." 

"  Fractions  are  parts  of  an  integer."  second. 

"  When   a   number  or  thing   is   divided  into      Third. 
equal  parts,  these  parts  are  called  FRACTIONS." 

Now,  will  either  of  these  definitions  convey 
to  the  mind  of  a  learner,  a  distinct  and  exact 
idea  of  a  fraction  ? 

The  term  "fraction,"  as  used  in  Arithmetic,  Term  fraction 

,  ,,  .  .  defined. 

means  one  or  more  equal   parts   ol   something 
regarded  as  a  whole  :  the  parts  to  be  expressed 
in  terms  of  the  thing  divided  CONSIDERED  AS  A 
UNIT.     There  are  three  prominent  ideas  which      ideas 
the  mind  must  embrace  : 

1st.  That  the  thing  divided  be  regarded  as  a      First. 
standard,  or  unity  ; 

2d.  That  it  be  divided  into  equal  parts  ;  second. 

3d.  That  the  parts  be  expressed  in  terms  of     Third. 
the  thing  divided,  regarded  as  a  unit. 

These  ideas  are  referred  to  in  the  latter  part 


of  the   first   definition.      Indeed,   the   definition       ^ed: 
would  suggest  them  to  any  one  acquainted  with 
the  subject,  but  not,  we  think,  to  a  learner. 

In  the  second  definition,  neither  of  them  is     isafrac- 

...  _.  11-  ti°n  P81*  °f 

hinted  at.     Take,  for  example,  the  integer  num-    ^  integer? 

ber  12,  and  no  one  would  say  that  any  one  part 
of  this  number,  as  2,  4,  or  6,  is  a  fraction. 


198  MATHEMATICAL     SCIENCE.  [BOOK  II. 


Third  The  third  definition  would  be  perfectly  accu- 

'   rate,  by  inserting  after  the  word   "thing,"  the 

words,  "  regarded  as  a  whole."     It  very  clearly 

expresses  the  idea  of  equal  parts,  but  does  not 

in  what  de-   present  the  idea  strongly  enough,  that  the  thing 

divided  must  be  regarded  as  unity,  and  that  the 

parts  must  be  expressed  in  terms  of  this  unity. 

§  205.  I  have  thus  given  a  few  examples,  illus- 
Necessity  of  trating  the  necessity  of  accurate  definitions  and 
terms.     Nothing  further  need  be  added,  except 
the  remark,  that  terms  should  always  be  used  in 
the  same  sense,  precisely,  in  which  they  are  de 
fined, 
objection         To  some,  perhaps,  these  distinctions  may  ap- 

to  exactness 

of  thought  Pear  over-nice,  and  matters  of  little  moment. 
e'  It  may  be  supposed  that  a  general  impression, 
imparted  by  a  language  reasonably  accurate, 
will  suffice  very  well ;  and  that  it  is  hardly 
worth  while  to  pause  and  weigh  words  on  a 
nicely-adjusted  balance. 

Any  such  notions,  permit  me  to  say,  will  lead 
to  fatal  errors  in  education. 
Definitions  m      It  is  in  mathematical  science  alone  that  words 

mathematics. 

are  the  signs  of  exact  and  clearly-defined  ideas. 
It  is  here  only  that  we  can  see,  as  it  were,  the 
very  thoughts  through  the  transparent  words  by 
which  they  are  expressed.  If  the  words  of  the 


CHAP.   II.]  ARITHMETIC SUBJECTS.  199 


definitions  are  not  such  as  convey  to  the  mind     Must  be 
of   the   learner,    the   fundamental   ideas   of   the   reason  cor- 
science,   he    cannot    reason   upon    these   ideas ; 
for,  he  does  not  apprehend  them ;  and  the  great 
reasoning  faculty,  by  which  all  the  subsequent 
principles  of  mathematics  are  developed,  is  en 
tirely  unexercised.* 

It  is   not  possible  to  cultivate  the   habit   of  cannot  other 
wise  cultivate 

accurate  thinking,  without  the  aid  and  use  of     habits  of 
exact  language.     No  mental  habit  is  more  use 
ful  than  that  of  tracing  out  the  connection  be 
tween  ideas  and  language.     In  Arithmetic,  that 
connection   can    be    made    strikingly   apparent,    connection 
Clear,    distinct    ideas — diamond    thoughts — may    W(misand 
be  strung  through  the  mind  on  the  thread  of 
science,  and  each  have  its  word  or  phrase  by 
which   it   can   be   transferred  to  the  minds  of 
others. 


HOW    SHOULD    THE    SUBJECTS    BE    PRESENTED? 

§  206.  Having  considered  the  natural  connec-      what 

has  been 

tion   of  the   subjects   of  arithmetic   with   each   considered, 
other,  as  branches  of  a  single  science,  based  on 
a  single  unit ;    and   having   also   explained   the 
necessity   of   a   perspicuous    and   accurate  lan- 


*  Section  200. 


200  MATHEMATICAL     SCIENCE.  [BOOK  II 


HOW  ought   guage  ;  we  come  now  to  that  important  inquiry, 

the  subjects 

tobepre-     How  ought  those  subjects  to  be  presented  to  the 
mind  of  a  learner  ?    Before  answering  this  ques- 
TWO  objects  tion,  we  should  reflect,  that  two  important  ob- 
jects  should  be  sought  after  in  the  study  of  arith 


metic  : 

First.  1st.    To  train    the   mind   to  habits   of  clear, 

quick,    and    accurate    thought  —  to    teach    it   to 

apprehend  distinctly  —  to  discriminate  closely  — 

to  judge  truly  —  and  to  reason  correctly  ;  and, 

second.         2d.    To   give,    in    abundance,    that   practical 

knowledge  of  the  use  of  figures,   in   their  va 

rious  applications,  which  shall  illustrate  the  stri- 

Artofarith-  king  fact,  that  the  art  of  arithmetic  is  the  most 

metic. 

important  art  of  civilized  life  —  being,   in    fact, 
the  foundation  of  nearly  all  the  others. 

now  first  im-      §  207.   It  is  certainly  true,  that  most,  if  not 

prussions  are 

made.  all  the  elementary  notions,  whether  abstract  or 
practical  —  that  is,  whether  they  relate  to  the 
science  or  to  the  art  of  arithmetic,  must  be 
made  on  the  mind  by  means  of  sensible  objects. 
Because  of  this  fact,  many  have  supposed  that 
is  reason-  the  processes  of  reasoning  are  all  to  be  con 

ing  to  be  con-    ,  11 

ducted  by    ducted  by  the  same  sensible  objects  ;    and  that 

objects?     every  abstract  principle  of  science  is  to  be  de 

veloped    and    established    by    means    of    sofas, 

chairs,  apples,  and  horses.     There  seems   to  be 


CHAP.   II.]  ARITHMETIC SUBJECTS.  201 


an   impression   that   because   blocks   are   useful     sensible 

.  ,  ,  .  .  TIT  7          /.          objects  useful 

aids   in   teaching   the    alphabet,    that,    therefore  m  acquiring 
they   can   be   used    advantageously   in    reading  th*9imPlest 

J  J  &      elements : 

Milton  and  Shakspeare.     This  error  is  akin  to 
that  of  attempting  to  teach   practically,  Geog 
raphy  and  Surveying  in  connection  with  Geom-      Error 
etry,  by  calling  the  angles  of  a  rectangle,  north,   °  1^"* 
south,  east,  and  west,  instead  of  simply  designa 
ting  them  by  the  letters  A,  B,  C,  and  D. 

This  false  idea,  that  every  principle  of  sci-   False  idea: 
ence    must   be    learned  practically,    instead   of 
being  rendered  practical  by  its  applications,  has    its  effects. 
been  highly  detrimental  both  to  science  and  art. 

A  mechanic,  for  example,  knowing  the  height     Example 
of  his  roof  and  the  width  of  his  building,  wishes     cation  of 
to  cut  his  rafters  to  the  proper  length.     If  he   ^^ 
calls  to  his  aid  the  established,  though  abstract 
principles  of  science,  he  finds  the  length  of  his 
rafter,  by  the  well-known  relation  between  the 
hypothenuse  and  the  two  sides  of  a  right-angled 
triangle.     If,  however,  he  will  learn  nothing  ex 
cept  practically,  he  must  raise  his  rafter  to  the   or  learning 

practically. 

roof,  measure  it,  and  if  it  be  too  long  cut  it  on, 
if  too  short,  splice  it.  This  is  the  practical  way 
of  learning  things. 

The  truly  practical  way,  is  that  in  which  skill       Tme 

practical. 

is  guided  by  science. 

Do  the  principles  above  stated  find  any  appli- 


202  MATHEMATICAL     SCIENCE.  [BOOK   II. 

cation  in  considering  the  question,  How  should 
can       arithmetic   be  taught?     Certainly  they  do.     If 

be  applied.  .  -, 

arithmetic    be    both   a   science    and   an   art,   it 
should  be  so  taught  and  so  learned. 


Principles  §  208.  The  principles  of  every  science  are  gen 
eral  and  abstract  truths.  They  are  mere  ideas, 

what  primarily  acquired  through  the  senses  by  experi 
ence,  and  generalized  by  processes  of  reflection 

wise      and  reasoning;  and  when  understood,  are  certain 

t;>  use  them.          .  i  •    i      i  TIT 

guides  in  every  case  to  which  they  are  applicable, 
If  we  choose  to  do  without  them,  wre  may.  But 
is  it  wise  to  turn  our  heads  from  the  guide-boards 
and  explore  every  road  that  opens  before  us  ? 

Now,  in  the  study  of  arithmetic  those  princi 
ples  of  science,  applicable  to  classes   of  cases, 
when      should  always  be  taught  at  the  earliest  possible 

and  how 

they  should  moment.      The   mind    should    never   be   forced 

through  a  long  series  of  examples,  without  ex- 

The  methods  planation.     One  or  two  examples  should  always 

pointed  out. 

precede  the  statement  of  an  abstract  principle, 
or  the  laying  down  of  a  rule,  so  as  to  make  the 
language  of  the  principle  or  rule  intelligible. 
But  to  carry  the  learner  forward  through  a 
Principles  series  of  them,  before  the  principle  on  which 

to  be  impres 
sed,       they  depend  has  been  examined  and  stated,  is 

forcing  the  mind  to  advance  mechanically — it 
is  lifting  up  the  rafter  to  measure  it,  when  its 


CHAP.  II.]  ARITHMETIC TEXT-BOOKS.  203 

exact  length  could  be  easily  determined  by  a 
mle  of  science. 

As  most  of  the  instruction  in  arithmetic  must      Books: 
be  given  with  the  aid  of  books,  we  feel  unable 
to  do  justice  to  this  branch  of  the  subject  with-    Necessity 

,  for  treating 

out  submitting  a  few  observations  on  the  nature     of  them, 
of  text-books  and  the  objects  which  they  are  in 
tended  to  answer. 


TEXT-BOOKS. 

§  209.   A  text-book  should  be  an  aid  to  the   Text-book: 
teacher   in    imparting   instruction,    and   to    the 
learner  in  acquiring  knowledge. 

It  should  present   the  subjects  of  knowledge     whatu 

.  ,,  should  be. 

in  their  proper  order,  with  the  branches  ol  each 
subject  classified,  and  the  parts  rightly  arranged. 
No  text-book,  on  a  subject  of  general  knowledge,  selection 

.  ~  .  of  subjects 

can  contain  all  that  is  known  ol  the  subject  on    necessary. 

which  it  treats ;  and  ordinarily,  it  can  contain 

but  a  very  small  part.     Hence,  the  subjects  to 

be  presented,  and  the  extent  to  which  they  are    Difficulties 

,  ,  „.,....          of  selection. 

to  be  treated,  are  matters  of  nice  discrimination 
and  judgment,  about  which  there  must  always 
be  a  diversity  of  opinion. 

§  210.  The  subjects  selected  should  be  leading    subjects: 
ones,  and  those  best   calculated   to  unfold,  ex- 


204  MATHEMATICAL     SCIENCE.  [flOOK  II. 


plain,  and  illustrate  the  principles  of  the  science. 
HOW       They  should  be  so  presented  as  to  lead  the  mind 

presented.  .        .      . 

to  analyze,  discriminate,  and  classify ;  to  see 
each  principle  separately,  each  in  its  combina 
tion  with  others,  and  all,  as  forming  an  harmo 
nious  whole.  Too  much  care  cannot  be  be- 
suggestive  stowed  in  forming  the  suggestive  method  of 

method : 

arrangement :    that   is,   to  place    the  ideas    and 
principles  in  such  a  connection,  that  each  step 
Reason  for.   shall  prepare  the  mind  of  the  learner  for  the  next 
in  order. 


object          §  211.  A  text-book  should  be  constructed  for 

of  a  text- 

book:  the  purpose  of  furnishing  the  learner  with  the 
keys  of  knowledge.  It  should  point  out,  explain, 
Nature;  and  illustrate  by  examples,  the  methods  of  in 
vestigating  and  examining  subjects,  but  should 
leave  the  mind  of  the  learner  free  from  the  re- 
straints  of  minute  detail.  To  fill  a  book  with 
the  analysis  of  simple  questions,  which  any  child 
can  solve  in  his  own  way,  is  to  constrain  and 
force  the  mind  at  the  very  point  where  it  is  ca 
pable  of  self-action.  To  do  that  for  a  pupil, 
which  he  can  do  for  himself,  is  most  unwise. 


detail 


*  212<    A   text-book  on  a  subject  of  science 
toricai.      should  not  be  historical.     At  first,  the  minds  of 
children  are  averse  to  whatever  is  abstract,  be- 


CHAP.   II.]  ARITHMETIC TEXT -BOOKS.  205 

cause  what  is  abstract  demands  thought,  and  Reasons, 
thinking  is  mental  labor  from  which  untrained 
minds  turn  away.  If  the  thread  of  science  be 
broken  by  the  presentation  of  facts,  having  no 
connection  with  the  argument,  the  mind  will 
leave  the  more  rugged  path  of  the  reasoning, 
and  employ  itself  with  what  requires  less  effort 
and  labor. 

The  optician,  in  his  delicate  experiments,  ex-  illustration, 
eludes  all  light  except  the  beam  which  he  uses : 
so,    the    skilful    teacher    excludes    all    thoughts 
excepting   those  which  he   is   most  anxious  to 
impress. 

As  a  general  rule,  subject  of  course  to  some 
exceptions,    but   one   method   for   each   process  one  method, 
should  be  given.     The  minds  of  learners  should 
not  be  confused.     If  several  methods  are  given,     Reasons, 
it  becomes  difficult  to  distinguish  the  reasonings 
applicable  to  each,  and  it  requires  much  knowl 
edge  of  a  subject  to  compare  different  methods 
with  each  other. 


§  213.  It  seems  to  be  a  settled  opinion,  both     HOW  the 

subject  is 

among  authors  and  teachers,  that  the  subject  of     divided, 
arithmetic  can  be  best  presented  by  means  of 
three  separate  works.     For  the  sake  of  distinc 
tion,  we  will  designate  them  the  First,  Second, 
and  Third  Arithmetics. 


206  MATHEMATICAL     SCIENCE.  [BOOK    II. 


We  will  now  explain  what  we  suppose  to  be 
the  proper  construction  of  each  book,  and  the 
object  for  which  each  should  be  designed. 


FIRST     ARITHMETIC. 

First  §  214.    This  book  should  give   to   the   mind 

Arithmetic : 

its  first  direction  in  mathematical  science,  and 

its   first    impulse    in    intellectual    development. 

its        Hence,  it   is  the  most   important   book  of  the 

importance. 

series.     Here,  the  faculties  of  apprehension,  dis 
crimination,  abstraction,  classification  and  com 
parison,  are  brought  first  into  activity.     Now, 
HOW       to  cultivate  and  develop  these  faculties  rightly, 

the  subjects 

must  be  we  must,  at  first,  present  every  new  idea  by 
means  of  a  sensible  object,  and  then  immedi 
ately  drop  the  object  and  pass  to  the  abstract 
thought. 

order          We  must  also  present  the  ideas  consecutively; 

of  the  ideas.  .... 

that  is,  in  their  proper  order ;  and  by  the  mere 
method  of  presentation  awaken  the  comparative 
and  reasoning  faculties.  Hence,  every  lesson 
should  contain  a  given  number  of  ideas.  The 
construction  ideas  of  each  lesson,  beginning  with  the  first, 

of  the  lessons.     . 

should  advance  in  regular  gradation,  and  the 
lessons  themselves  should  be  regular  steps  in 
the  progress  and  development  of  the  arithmeti 
cal  science. 


CHAP.  II.]  ARITHMETIC TEXT- BOOKS.  207 

6  215.  The  first  lesson  should  merely  contain       First 

lesson. 

representations  of  sensible  objects,  placed  oppo 
site  names  of  numbers,  to  give  the  impression 
of  the  meanings  of  these  names :  thus, 

One *  Whatit 

should  con- 
TWO *  *  tain. 

Three *  *  * 

&c.  &c. 

And  with  young  pupils,  more  striking  objects 
should  be  substituted  for  the  stars. 

In  the  second  lesson,  the  words  should  be  re 
placed  by  the  figures  :  thus, 

1 * 

2---------          •*  *•  Second 

3 #** 

&c.  &c. 

In  the  third  lesson,  I  would  combine  the  ideas 
of  the  first  two,  by  placing  the  words  and  fig 
ures  opposite  each  other :  thus, 


One  -  -  -  -  1 
Two  ....  2 
Three  -  -  -  3 
&c.  &c. 


The  Roman  method  of  representing  numbers 
should  next  be  taught,  making  the  fourth  lesson : 
viz., 


Four  -.--     4 

Five  -     ---     5          Third 

Six     -     -     -     -     6 

&c.  &c. 


lesson. 


208 


MATHEMATICAL     SCIENCE 


[BOOK 


Fourth 
lesson. 

One  - 
Two  - 

-     -     -     I. 
-     -    -  II. 

Four    -     - 
Five    -     - 

-     IV. 
-      V. 

Roman 
method. 

Three 
&c. 

-     -       III. 
&c. 

Six      -     - 
&c. 

-    VI. 

&c. 

First 
ten  combi- 

§  216. 

We  come  now  to  the  first 

ten  cor 

binations  of  numbers,  which  should  be  given  in 
a  separate  lesson.  In  teaching  them,  we  must, 
of  course,  have  the  aid  of  sensible  objects.  We 
teach  them  thus  : 

One        and         one         are  how  many  ? 


How 

taught  by 
things : 


* 
One 

* 
One 

* 
&c. 


and         two         are  how  many  ? 

*  * 

and        three        are  how  many  ? 
*  *  * 
&c.  &c., 


How  in 
the  abstract. 


through  all  the  combinations :  after  which,  we 
pass  to  the  abstract  combinations,  and  ask,  one 
and  one  are  how  many  ?  one  and  two,  how 
many  ?  one  and  three,  &c. ;  after  which  we 
express  the  results  in  figures. 

We  would  then  teach  in  the  same  manner,  in 

second      a  separate  lesson,  the  second  ten  combinations ; 

tons.      then  the  third,  fourth,  fifth,  sixth,  seventh,  eighth, 

ninth,  and  tenth.     In  the  teaching  of  these  com- 

wordsused.  binations,  only  the  words  from  one  to  twenty 

will  have  been  used.     We  must  then  teach  the 


CHAP.   II.]  ARITHMETIC TEXT-BOOKS. 


209 


combinations  of  which  the  results  are  expressed     Further 

,          ,  ,  combina- 

by  the  words  from  twenty  to  one  hundred.  tiong€ 


Results. 


How 
they  appear. 


§  217.  Having  done  this,  in  the  way  indi 
cated,  the  learner  sees  at  a  glance,  the  basis  on 
which  the  system  of  common  numbers  is  con 
structed.  He  distinguishes  readily,  the  unit  one 
from  the  unit  ten,  apprehends  clearly  how  the 
second  is  derived  from  the  first,  and  by  com 
paring  them  together,  comprehends  their  mutual 
relation. 

Having  sufficiently  impressed  on  the  mind  of 
the  learner,  the  important  fact,  that  numbers  are 
but  expressions  for  one  or  more  things   of  the 
same  kind,  the  unit  mark  may  be  omitted  in  the    Unit  mark 
combinations  which  follow. 


same 


§  218.  With  the  single  difference  of  the  omis- 
sion  of  the  unit  mark,  the  very  same  method 
should  be  used  in  teaching  the  one  hundred  ruje8* 
combinations  in  subtraction,  the  one  hundred 
and  forty-four  in  multiplication,  and  the  one 
hundred  and  forty-four  in  division. 

When  the  elementary  combinations  of  the  four 
ground  rules  are  thus  taught,  the  learner  looks    Results  of 

,        ,       ,  ,  •  r  i  •  the  method: 

back  through  a  series  of  regular  progression,  in 
which  every  lesson  forms  an  advancing  step, 
and  where  all  the  ideas  of  each  lesson  have  a 

14 


210  MATHEMATICAL     SCIENCE.  [BOOK  II. 


mutual  and  intimate  connection  with  each  other. 
Are  they     Will  not  such  a  system  of  teaching  train   the 

desirable?          .  r  ,.  i      •  i 

mind  to  the  habit  01  regarding  each  idea  sepa- 
The  rately — of  tracing  the  connection  between  each 
give.  new  idea  and  those  previously  acquired — and  of 

comparing  thoughts  with  each  other  ? — and  are 

not  these  among  the  great  ends  to  be  attained, 

by  instruction  ? 


§  219.  It  has  seemed  to  me  of  great  import- 
Figures     ance  to  use  figures  in  the  very  first  exercises  of 

should  be  .   ,  .  rT  . 

used  early,  arithmetic.  Unless  this  be  done,  the  operations 
must  all  be  conducted  by  means  of  sounds,  and 

Reasons,     the  pupil  is  thus  taught  to  regard  sounds  as  the 

proper   symbols    of   the    arithmetical    language. 

conse-      This    habit   of  mind,  once  firmly  fixed,  cannot 

quences  of 

using  words  be  easily  eradicated;  and  when  the  figures  are 
learned  afterwards,  they  will  not  be  regarded 
as  the  representatives  of  as  many  things  as 
their  names  respectively  import,  but  as  the  rep 
resentatives  merely  of  familiar  sounds  which 
have  been  before  learned. 

This   would   seem    to   account   for   the   fact, 
about  which,  I  believe,  there  is  no  difference  of 

oral       opinion ;   that  a  course  of  oral  arithmetic,  ex- 
arithmetic:         1.  .        , 

tending  over  the  whole  subject,  without  the  aid 

and  use  of  figures,  is  but   a   poor   preparation 
for  operations  on  the  s^ate.     It  may,  it  is  true, 


CHAP.  II.]  ARITHMETIC TEXT-BOOKS.  211 


sharpen  and  strengthen  the  mind,   and  give  it      what 

it  may  do. 

development:  but  does  it  give  it  that  language 

and  those  habits  of  thought,  which  turn  it  into     what  it 

does  not  do. 

the  pathways  of  science?  The  language  of  a 
science  affords  the  tools  by  which  the  mind  Language 

.        ,  .  ,  ,         of  arithmetic: 

pries  into  its  mysteries  and  digs  up  its  hidden 
treasures.     The  language  of  arithmetic  is  formed 
from  the  ten  figures.     By  the  aid  of  this  Ian-     its  uses, 
guage  we  measure  the  diameter  of  a  spider's 
web,   or    the    distance   to   the   remotest   planet      what 
which  circles  the  heavens  ;   by  its  aid,  we  cal 
culate  the  size  of  a  grain  of  sand  and  the  mag 
nitude  of  the  sun  himself:  should  we  then  aban 
don  a  language  so  potent,  and  attempt  to  teach     its  value, 
arithmetic   in    one   which   is    unknown    in    the 
higher  departments  of  the  science  ? 

§  220.   We  next  come  to  the  question,  how    Fractions: 
the  subject  of  fractions  should  be  presented  in 
an  elementary  work. 

The  simplest  idea  of  a  fraction  comes  from     simplest 

idea- 

dividing  the  unit  one  into  two  equal  parts.     To 
ascertain  if  this  idea  is  clearly  apprehended,  put       HOW 
the  question,   How  many  halves   are   there   in 
one  ?     The  next  question,  and  it  is  an  import-       Next 

TT  .          question. 

ant  one,  is  this :  How  many  halves  are  there  in 
one  and  one-half?  The  next,  How  many  halves 
in  two  ?  How  many  in  two  and  a  half  ?  In 


212  MATHEMATICAL     SCIENCE.  [BOOK  II 

three  ?  Three  and  a  half?  and  so  on  to  twelve. 
Results.  You  will  thus  evolve  all  the  halves  from  the 
units  of  the  numbers  from  one  to  twelve,  in 
clusive.  We  stop  here,  because  the  multipli 
cation  table  goes  no  further.  These  combina- 

First  lesson,  tions  should  be  embraced  in  the  first  lesson  on 
fractions.  That  lesson,  therefore,  will  teach  the 

rts  extent,  relation  between  the  unit  1  and  the  halves,  and 
point  out  how  the  latter  are  obtained  from  the 
former. 

second  §  221.  The  second  lesson  should  be  the  first, 
reversed.  The  first  question  is,  how  many 

Grades  whole  things  are  there  in  two  halves  ?  Sec 
ond,  How  many  whole  things  in  four  halves  ? 
How  many  in  eight  ?  and  so  on  to  twenty-four 
halves,  when  we  reach  the  extent  of  the  division 
Extent  of  table.  In  this  lesson  you  will  have  taught  the 
pupil  to  pass  back  from  the  fractions  to  the  unit 
from  which  they  are  derived. 

Fundamental  §  222.  You  have  thus  taught  the  two  funda 
mental  principles  of  all  the  operations  in  frac 
tions:  viz. 

First.  1st.  To  deduce  the  fractional  units  from  in 

teger  units ;  and, 

second.  2dly.  To  deduce  integer  units  from  fractional 
units. 


CHAP 


.  II.]  ARITHMETIC TEXT-BOOKS.  213 


§  223.  The  next  lesson  should  explain  the  law 
by  which  the  thirds  are  derived  from  the  units      thirds. 
from  1  to  12  inclusive  ;  and  the  following  lesson 
the  manner  of  changing  the  thirds  into  integer 
units. 

The  next  two  lessons  should  exhibit  the  same     Fourths 

and  other 

operations   performed   on   the   fourth,   the   next     fractions. 
two   on   the   fifth,   and   so   on    to    include    the 
twelfth. 


§  224.  This  method  of  treating  the  subject  of  Advantages 

of  the 

fractions  has  many  advantages  :  method. 

1st.  It  points  out,  most  distinctly,  the  relations 
between  the  unit  1  and  the  fractions  which  are       First. 
derived  from  it. 

2d.  It  points  out  clearly  the  methods  of  pass-     second. 
ing  from  the  fractional  to  the  integer  units. 

3d.    It  teaches  the  pupil  to  handle  and  com-       Third. 
bine  the  fractional  units,  as  entire  things. 

4th.  It  reviews  the  pupil,  thoroughly,  through     Fourth. 
the  multiplication  and  division  tables. 

5th.    It  awakens  and  stimulates  the  faculties      Fifth. 
of  apprehension,  comparison,  and  classification. 

§  225.    Besides   the   subjects   already  named,      what 

T-«-  •  '1111  -1  6lSe   the  Fil'St 

the   rirst   Arithmetic    should    also   contain   the    Arithmetic 
tables   of  denominate   numbers,  and  collections 
of  simple  examples,  to  be  worked  on  the  slate, 


214 


MATHEMATICAL     SCIENCE.  [fiOOKII. 


Examples,    under  the  direction  of  the  teacher.     It  is  not 

how  taught. 

supposed  that  the  mind  of  the  pupil  is  suffi 
ciently  matured  at  this  stage  of  his  progress  to 
understand  and  work  by  rules. 


what 

should  be 

taught  in 


second. 

Third. 
Fourth. 
Fifth. 


§  226.    In    the    First   Arithmetic,    therefore, 

,  -i      i         i  T    i 

tne  PUP"  should  be  taught, 

1st.  The  language  of  figures; 

2d.  The  four  hundred  and  eighty-eight  ele- 
mentary  combinations,  and  the  words  by  which 
they  are  expressed  ; 

3d.  The  main  principles  of  Fractions  ; 

4th.  The  tables  of  Denominate  Numbers  ;  and, 

5th.  To  perform,  upon  the  slate,  the  element 
ary  operations  in  the  four  ground  rules. 


second 

Arithmetic. 


whatu 


SECOND     ARITHMETIC. 

§  227.  This  arithmetic  occupies  a  large  space 

. 

in  the  school  education  of  the  country.  Many 
study  it,  who  study  no  other.  It  should,  there- 
fore,  be  complete  in  itself.  It  should  also  be 
eminently  practical;  but  it  cannot  be  made  so 
either  by  giving  it  the  name,  or  by  multiplying 
the  examples. 


Practical         §  228.  The  truly  practical  cannot  be  the  ante- 

application  of 

principle,    cedent,  but  must  be  the  consequent  of  science. 


CHAP.   II.]  ARITHMETIC TEXT-BOOKS.  215 


Hence,    that    general    arrangement    of   subjects  Arrangement 
demanded   by   science,    and    already   explained, 
must  be  rigorously  followed. 

But  in  the   treatment  of  the  subjects  them-  Reasons  for 

.  departures. 

selves,  we  are  obliged,  on  account  of  the  limited 
information  of  the  learners,  to  adopt  methods  of 
teaching  less  general  than  we  could  desire. 


§  229.   We  must  here,  again,  begin  with  the       Basis- 
unit  one,  and  explain  the  general  formation  of 
the    arithmetical   language,   and   must    also   ad 
here  rigidly  to  the  method  of  introducing  new     Method, 
principles  or  rules  by  means  of  sensible  objects. 

This  is  most  easily  and  successfully  done  either       How- 
carried  out. 
by  an  example  or  question,  so  constructed  as  to 

show  the  application  of  the  principle  or  rule. 
Such  questions  or  examples  being  used  merely 
for  the  purpose  of  illustration,  one  or  two  will  Few 

examples. 

answer  the  purpose  much  better  than  twenty  : 

for,  if  a  large    number  be  employed,  they  are     Reasons. 

regarded  as  examples  for  practice,  and  are  lost 

sight   of  as   illustrations.     Besides,    it   confuses 

the   mind  to  drag  it  through   a  long  series  of 

examples,   before   explaining    the   principles   by 

which  they  are  solved.     One  example,  wrought  one  example 

under  a  rule. 

under  a  principle  or  rule  clearly  apprehended, 
conveys  to  the  mind  more  practical  informa 
tion,  than  a  dozen  wrought  out  as  independent 


216 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


Principle,    exercises.     Let  the  principle  precede  the  prac- 
Practice.     tice,   in    all    cases,   as    soon    as   the  information 

acquired  will  permit.     This  is  the  golden  rule 

both  of  art  and  morals. 


subjects         §  230.    The    Second   Arithmetic    should   em- 
embraced.    .  ,  .  .-.  „ 

brace  all  the  subjects  necessary  to  a  lull  view 

of  the  science  of  numbers  ;  and  should  contain 
an  abundance  of  examples  to  illustrate  their 
practical  applications.  The  reading  of  numbers, 
so  much  (though  not  too  much)  dwelt  upon,  is 
an  invaluable  aid  in  all  practical  operations. 

By  its  aid,  in  addition,  the  eye  runs  up  the 
columns  and  collects,  in  a  moment,  the  sum  of 
subtraction:  all  the  numbers.  In  subtraction,  it  glances  at 
the  figures,  and  the  result  is  immediately  sug 
gested.  In  multiplication,  also,  the  sight  of  the 
figures  brings  to  mind  the  result,  and  it  is 
reached  and  expressed  by  one  word  instead  of 
five.  In  short  division,  likewise,  there  is  a  cor 
responding  saving  of  time  by  reading  the  results 
of  the  operations  instead  of  spelling  them.  The 
method  of  reading  should,  therefore,  be  con 
stantly  practised,  and  none  other  allowed. 


Reading : 


Its  value 
in  Addition 


Multi 
plication  ; 


Division. 


CHAP.   II.]  ARITHMETIC TEXT-BOOKS.  217 


THIRD    ARITHMETIC. 

§  231.  We  have  now  reached  the  place  where      Third 

.  ,  .  -T,,         Arithmetic: 

arithmetic   may  be  taught  as  a  science.     The 
pupil,  before  entering  on  the  subject  as  treated  Preparation 
here,  should  be  able  to  perform,  at  least  mechan 
ically,  the  operations  of  the  five  ground  rules. 

Arithmetic  is  now  to  be  looked  at  from  an 
entirely   different    point   of   view.      The   great   view  of  it. 
principles  of  generalization  are  now  to  be  ex 
plained  and  applied. 

Primarily,   the    general    language    of    figures      what 

.,  .  is  taught 

must  be  taught,  and  the  striking  tact  must  then    primarily. 

be  explained,  that  the  construction  of  all  integer 

numbers   involves    but   a   single   principle,   viz. 

the  law  of  change  in  passing  from  one  unit  to  General  law: 

another.     The  basis  of  all  subsequent  operations 

will  thus  have  been  laid. 

§  232.  Taking  advantage  of  this  general  law 
which  controls  the   formation    of  numbers,  we     controls 

.  r  formation  of 

bring  all  the  operations  01  reduction  under  one    numbers 
single  principle,  viz.  this  law  of  change  in  the 
unities. 

Passing  to  addition,  we  are  equally  surprised     its  value 
and  delighted  to  find  the   same  principle  con 
trolling  all  its  operations,  and  that  integer  num 
bers  of  all  kinds,  whether  simple  or  denominate, 
may  be  added  under  a  single  rule. 


218  MATHEMATICAL     SCIENCE.  [BOOK   II. 

Advantages       This  view  opens  to  the  mind  of  the  pupil  a 

general  law.  wide  field  of  thought.     It  is   the   first  illustra 

tion  of  the  great  advantage  which  arises  from 

looking  into  the   laws   by  which   numbers    are 

subtraction,  constructed.      In    subtraction,    also,    the    same 

principle  finds  a  similar  application,  and  a  sim 

ple  rule  containing  but  a  few  words  is  found 

applicable  to  all  the  classes  of  integer  numbers. 

In  multiplication  and  division,  the  same  stri 

king   results   flow   from  the    same    cause;    and 

General     thus  this  simple  principle,  viz.  the  law  of  change 

law  of  num-     .  ,,  •         /•         7  7  • 

bers:  in  passing  jrom  one  unit  oj  value  to  another,  is 
the  key  to  all  the  operations  in  the  four  ground 
rules,  whether  performed  on  simple  or  denomi 
nate  numbers.  Thus,  all  the  elementary  opera- 
controis  tions  of  arithmetic  are  linked  to  a  single  prin- 
ciple>  and  that  one  a  mere  principle  of  arith 
metical  language.  Who  can  calculate  the  la 
bor,  intellectual  and  mechanical,  which  may 
be  saved  by  a  right  application  of  this  lumin 
ous  principle  ? 


Design  §  233.  It  should  be  the  design  of  a  higher 
°LuhmeticT  arithmetic  to  expand  the  mind  of  the  learner 
over  the  whole  science  of  numbers  ;  to  illus 
trate  the  most  important  applications,  and  to 
make  manifest  the  connection  between  the  sci 
ence  and  the  art. 


CHAP.   II.J  ARITHMETIC TEXT-BOOKS.  219 

It  will  not  answer  these  objects  if  the  methods        its 

requisites. 

of  treating  the  subject  are  the  same  as  in  the 
elementary  works,  where  science  has  to  com 
promise  with  a  want  of  intelligence.  An  ele 
mentary  is  not  made  a  higher  arithmetic,  by  Must  have 

.          .         -i    r    •   •  .  ..,          a  distinctive 

merely  transferring  its  definitions,  its  principles,    character, 
and   its  rules  into  a  larger  book,  in  the  same 
order  and  connection,  and  arranging  under  them 
an  apparently  new  set  of  examples,  though  in  fact 
constructed  on  precisely  the  same  principles. 

§  234.    In  the  four  ground  rules,  particularly     construc- 

.  tion  of  exain- 

(where,  in  the  elementary  works,  simple  exam-  P]esinthe 
pies  must  necessarily  be  given,  because  here 
they  are  used  both  for  illustration  and  practice), 
the  examples  should  take  a  wide  range,  and  be 
so  selected  and  combined  as  to  show  thei^  com 
mon  dependence  on  the  same  principle. 


§  235.  It  being  the  leading  design  of  ft  series      Design 

.  .  i      ii  .       of  a  series. 

of  arithmetics  to  explain  Hud  JIustrate  the  sci 
ence  and  art  of  numbers,  great  care  should  be 
taken   to  treat  all  the   subjects,  as   far  as   their 
different  natures  will  permit,  according  to  the 
same   general   methods.      In   passing   from   one 
book  to  another,  every  subject  which  has  been     subjects 
fully  and  satisfactorily  treated  in  the  one,  should  ferredwhen 
be  transferred  to  the  other  with  the  fewest  pos-  fully treated' 


220  MATHEMATICAL     SCIENCE.  [BOOK  II. 


HOW  com-    sible  alterations  ;  so  that  a  pupil  shall  not  have 

mon  subjects 

maybe  to  learn  under  a  new  dress  that  which  he  has 
already  fully  acquired.  They  who  have  studied 
the  elementary  work  should,  in  the  higher  one, 
either  omit  the  common  subjects  or  pass  them 
over  rapidly  in  review. 

The  more  enlarged  and  comprehensive  views 

Reasons,     which  should  be  given  in  the  higher  work  will 

thus  be  acquired  with  the  least  possible  labor,  and 

the  connection  of  the  series  clearly  pointed  out. 

This  use  of  those  subjects,  which  have  been 

fully  treated  in  the  elementary  work,  is  greatly 

preferable  to  the  method  of  attempting  to  teach 

Additional    every  thing  anew  :  for  there  must  necessarily  be 

stated,  much  that  is  common ;  and  that  which  teaches 
no  new  principle,  or  indicates  no  new  method  of 
application,  should  be  precisely  the  same  in  the 
higher  work  as  in  that  which  precedes  it. 

§  236.  To  vary  the  examples,  in  form,  without 

changing  in  the  least  the  principles  on  which 

A  contrary    they  are  worked,  and  to  arrange  a  thousand  such 

method  leads 

to  confusion:  collections  under  the  same  set  of  rules  and  sub 
ject  to  the  same  laws  of  solution,  may  give  a 
little  more  mechanical  facility  in  the  use  of 
figures,  but  will  add  nothing  to  the  stores  of 
arithmetical  knowledge.  Besides,  it  deludes  the 
learner  with  the  hope  of  advancement,  and  when 


CHAP.   II.]           ARITHMETIC  CONCLUSION. 

221 

he  reaches  the  end  of  his  higher  arithmetic,  he 
finds,  to  his  amazement,  that  he  has  been  con 

It  misleads 
the  pupil: 

ducted  by  the  same  guides  over  the  same  ground 

through    a    winding    and    devious    way,   made 
strange  by  fantastic  drapery  :   whereas,  if  what 

It  com 
plicates  the 
subject. 

was  new  had  been  classed  by  itself,  and  what 

was  known  clothed  in  its  familiar  dress,  the  sub 

ject  would  have  been  presented  in  an  entirely 

different  and  brighter  light. 

CONCLUDING     REMARKS. 

We  have  thus  completed  a  full  analysis  of  the 

Conclusion. 

language  of  figures,  and  of  the  construction  of 

numbers. 

We  have  traced  from  the  unit  one,   all  the 
numbers  of  arithmetic,  whether  integer  or  frac 

What 
has  been 
done. 

tional,  whether  simple  or  denominate.    We  have 

developed  the  laws  by  which  they  are  derived 

Laws. 

from    this   common   source,   and   perceived   the 

connections  of  each  class  with  all  the  others. 

We  have  examined  that  concise  and  beautiful 
language,  by  means  of  which  numbers  are  made 

Analysis 
of  the  lan 
guage. 

available   in   rendering   the    results    of    science 

practically  useful  ;  and  we  have  also  considered 
the  best  methods  of  teaching  this  great  subject 

Methods 
of  teaching 
indicated. 

—  the  foundation  of  all  mathematical  science  — 
and  the  first  among  the  useful  arts. 

Import 
ance  of  the 
subject. 

CSIAP.   III.]  GEOMETRY.  223 


CHAPTER    III. 

GEOMETRY    DEFINED THINGS     OF    "WHICH     IT     TREATS — COMPARISON    AND    PROP 
ERTIES     OF     FIGURES DEMONSTRATION PROPORTION SUGGESTIONS     FOR 

TEACHING. 

GEOMETRY. 

§  237.  GEOMETRY  treats  of  space,   and  com-    Geometry, 
pares  portions  of  space  with  each  other,  for  the 
purpose  of  pointing  out  their  properties  and  mu 
tual  relations.     The  science  consists  in  the  de-   its  science, 
velopment  of  all  the  laws  relating  to  space,  and 
is  made  up  of  the  processes  and  rules,  by  means 
of  which  portions  of  space  can  be  best  compared 
with  each  other.     The  truths  of  Geometry  are  a    Its  trutns- 
series  of  dependent  propositions,  and  may  be  di-    Of  three 
vided  into  three  classes  : 

1st.  Truths  implied  in  the  definitions,  viz.  that   1st.  Those 

implied  in 

things  do  exist,  or  may  exist,  corresponding  to    thedenni- 
the  words  defined.     For  example  :  when  we  say, 
"  A  quadrilateral  is  a  rectilinear  figure  having  four 
sides,"  we  imply  the  existence  of  such  a  figure. 

2d.  Self-evident,  or  intuitive  truths,  embodied  ^  A^oms. 
in  the  axioms ;  and, 

3d.  Truths  inferred  from  the  definitions  and  3d.  Demon- 


224  MATHEMATICAL     SCIENCE.  [BOOK  II. 

strative     axioms,  called  Demonstrative  Truths.     We  say 

truths.  ,  ... 

that  a  truth  or  proposition  is  proved  or  demon- 

When  de 
monstrated,   strated,   when,  by   a    course   of  reasoning,  it  is 

shown  to  be  included  under  some  other  truth  or 
proposition,  previously  known,  and  from  which 
is  said  to  follow ;  hence, 

Demonstrar          A  DEMONSTRATION    is    a    Series  of   logical  argU- 

ments.  brought  to  a  conclusion,  in  which  the 
major  premises  are  definitions,  axioms,  or  prop 
ositions  already  established. 

subjects  of       §  238.    Before  we  can  understand  the  proofs 

Geometry. 

or  demonstrations  of  Geometry,  we  must  under 
stand  what  that  is  to  which  demonstration  is 
applicable :  hence,  the  first  thing  necessary  is 
to  form  a  clear  conception  of  space,  the  subject 
of  all  geometrical  reasoning.* 

Names  of        The  next  step  is  to  give  names  to  particular 
ms'      forms  or  limited  portions  of  space,  and  to  define 
these  names  accurately.    The  definitions  of  these 
names  are  the  definitions  of  Geometry,  and  the 
portions    of   space   corresponding   to   them    are 
Figures,      called  Figures,  or  Geometrical  Magnitudes  ;    of 
Three  kinds,  which  there  are  three  general  classes : 
First.  1st.  Lines; 

second.         2d.  Surfaces ; 
Third.          3d.  Solids. 

*  Sections  81  to  85. 


CHAP.   III.] 


GEOMETRY 


2J5 


§  239.  Lines  embrace  only  one  dimension  of     Lines, 
space,  viz.  length,  without  breadth  or  thickness. 
The  extremities,  or  limits  of  a  line,  are   called 
points. 

There  are  two  general  classes  of  lines — straight  TWO  classes: 
lines  and  curved  lines.     A  straight  line  is  one     curred.. 
which  lies  in  the  same  direction  between    any 
two  of  its  points  ;  and  a  curved  line  is  one  which 
constantly  changes  its  direction  at  every  point. 
There  is  but  one  kind  of  straight  line,  and  that  is  one  kind  of 
fully  characterized  by  the  definition.     From  the 
definition  we  may  infer  the  following  axiom :  "  A 
straight  line  is  the  shortest  distance  between  two 
points."      There    are  many  kinds  of  curves,  of    many  of 
which  the  circumference  of  the  circle  is  the  sim 
plest  and  the  most  easily  described. 

§  240.    Surfaces  embrace  two  dimensions  of    surfaces: 
space,  viz.  length  and  breadth,  but  not  thickness. 
Surfaces,  like  lines,   are  also   divided  into   two    pianeand 
general  classes,  viz.  plane  surfaces  and  curved 
surfaces. 

A  plane  surface  is  that  with  which  a  straight     A  plane 
line,    any  how  placed,   and   having   two   points 
common  with  the  surface,  will  coincide  through 
out   its  entire  extent.     Such   a   surface  is   per 
fectly  even,  and  is  commonly  designated  by  the     Perfectly 

CY6U* 

term  "A  plane."     A  large  class  of  the  figures 

15 


226  MATHEMATICAL     SCIENCE.  [BOOK  II. 

piane  Fig-  of  Geometry  are  but  portions  of  a  plane,  and  all 
such  are  called  plane  figures. 

§  241.  A  portion  of  a  plane,  bounded  by  three 
A  triangle,    straight  lines,  is  called  a  triangle,  and  this  is  the 

the  most  sim 
ple  figure,    simplest  of  the  plane  figures.     There  are  several 

kinds  of  triangles,  differing  from  each  other, 
however,  only  in  the  relative  values  of  their 
sides  and  angles.  For  example :  when  the  sides 
are  all  equal  to  each  other,  the  triangle  is  called 
Kinds  of  tri-  equilateral ;  when  two  of  the  sides  are  equal,  it 
is  called  isosceles ;  and  scalene,  when  the  three 
sides  are  all  unequal.  If  one  of  the  angles  is  a 
right  angle,  the  triangle  is  called  a  right-angled 
triangle. 

§  242.  The  next  simplest  class  of  plane  figures 
comprises  all  those  which  are  bounded  by  four 

Quadriiater-  straight  lines,  and  are  called  quadrilaterals. 
There  are  several  varieties  of  this  class : 

1st  species.  3  st.  The  mere  quadrilateral,  which  has  no 
mark,  except  that  of  having  four  sides  ; 

2d  species.  2d.  The  trapezoid,  which  has  two  sides  par 
allel  and  two  not  parallel ; 

3d  species.  3d.  The  parallelogram,  which  has  its  opposite 
sides  parallel  and  its  angles  oblique ; 

4th  species.  4th.  The  rectangle,  which  has  all  its  angles 
right  angles  and  its  opposite  sides  parallel ;  and, 


CHAP.  III.]  GEOMETRY.  227 

5th.  The  square,  which  has  its  four  sides  equal   5th  species. 
to  each  other,  each  to  each,  and  its  four  angles 
right  angles. 

§  243.  Plane  figures,  bounded  by  straight  lines,  other  Poiy- 
having  a  number  of  sides  greater  than  four,  take 
names  corresponding  to  the  number  of  sides,  viz. 
Pentagons,  Hexagons,  Heptagons,  &c. 

§  244.    A   portion  of  a  plane  bounded  by  a     circles: 
curved  line,  all  the  points  of  which  are  equally 
distant  from  a  certain  point  within  called  the 
centre,  is  called  a  circle,  and  the  bounding  line 
is  called  the  circumference.     This  is  the  only  the  circum- 
curve  usually  treated  of  in  Elementary  Geometry. 

§  245.    A  curved  surface,  like  a  plane,  em-  curved  sur- 

faces: 

braces  the  two  dimensions  of  length  and  breadth. 
It  is  not  even,  like  the  plane,  throughout  its  whole 
extent,  and  therefore  a  straight  line  may  have  their  proper- 

ties 

two  points  in  common,  and  yet  not  coincide  with 
it.  The  surface  of  the  cone,  of  the  sphere,  and 
cylinder,  are  the  curved  surfaces  treated  of  in 
Elementary  Geometry. 

§  246.  A  solid  is  a  portion  of  space,  combi-      solids, 
ning  the  three  dimensions  of  length,  breadth,  and 
thickness.     Solids  are  divided  into  three  classes  :  Threeciasses. 


228  MATHEMATICAL     SCIENCE.  [BOOK   II. 

1st  class.         1st.  Those  bounded  by  planes  ; 
2d  class.         2d.  Those  bounded  by  plane  and  curved  sur 
faces  ;  and, 

3d  class.         3d.  Those  bounded  only  by  curved  surfaces, 
what  figures      The    first    class    embraces    the  pyramid   and 

fall  in  each          .  .  .        ,  . 

class.  prism  with  their  several  varieties  ;  the  second 
class  embraces  the  cylinder  and  cone ;  and  the 
third  class  the  sphere,  together  with  others  not 
generally  treated  of  in  Elementary  Geometry. 

Magnitudes  §  247.  We  have  now  named  all  the  geomet 
rical  magnitudes  treated  of  in  elementary  Ge- 

what  they  ometry.  They  are  merely  limited  portions  of 
space,  and  do  not,  necessarily,  involve  the  idea 

A  sphere,  of  matter.  A  sphere,  for  example,  fulfils  all  the 
conditions  imposed  by  its  definitions,  without  any 
reference  to  what  may  be  within  the  space  en- 
Need  not  be  closed  by  its  surface.  That  space  may  be  oc- 

material-  •  j  i,    i     j  •  •  i_ 

cupied  by  lead,  iron,  or  air,  or  may  be  a  vacuum, 

without  at  all  changing  the  nature  of  the  sphere, 
as  a  geometrical  magnitude. 

It  should  be  observed  that   the  boundary  or 

Boundaries  limit  of  a  geometrical  magnitude,  is  another  geo 
metrical  magnitude,  having  one  dimension  less. 
For  example :  the  boundary  or  limit  of  a  solid, 

Examples,  which  has  three  dimensions,  is  always  a  surface 
which  has  but  two :  the  limits  or  boundaries  of 


CHAP.   III.]  GEOMETRY.  229 

all  surfaces  are  lines,  straight  or  curved  ;  and  the 
extremities  or  limits  of  lines  are  points. 


§  248.    We  have  now  named  and  shown  the     subjects 

n  timed. 

nature  of  the  things  which  are  the  subjects  of 
Elementary  Geometry.  The  science  of  Ge-  Science°f 

Geometry. 

ometry  is  a  collection  of  those  connected  pro 
cesses  by  which  we  determine  the  measures, 
properties,  and  relations  of  these  magnitudes. 


COMPARISON  OF  FIGURES  WITH  UNITS  OF  MEASURE. 

§  249.  We  have  seen  that  the  term  measure     Measure, 
implies  a  comparison  of  the  thing  measured  with 
some  known  thing  of  the  same  kind,  regarded 
as  a  standard ;  and  that  such  standard  is  called 
the  unit  of  measure.*     The  unit  of  measure  for  unitofmeas- 
lines  must,  therefore,  be  a  line  of  a  known  length  :    For  Lines, 
a  foot,  a  yard,  a  rod,  a  mile,  or  any  other  known     A  Line' 
unit.     For  surfaces,   it  is  a  square  constructed  Forsurfaces, 
on  the  linear  unit  as  a  side :  that  is,  a  square   A  square. 
foot,  a  square  yard,  a  square  rod,  a  square  mile ; 
that  is,  a  square  described  on  any  known  unit 
of  length. 

The  unit  of  measure,  for  solidity,  is  a  solid,    ForSoiids, 
and  therefore  has  three  dimensions.     It  is  a  cube     A  cube. 

*  Section  94. 


230  MATHEMATICAL     SCIENCE.  [BOOK  II. 

constructed  on  a  linear  unit  as  an  edge,  or  on 
the  superficial  unit  as  a  base.     It  is,  therefore, 
a  cubic   foot,   a  cubic  yard,  a  cubic   rod,   &c. 
Three  units  Hence,   there  are  three  units  of  measure,  each 
differing  in  kind  from  the  other  two,  viz.  a  known 
A  Line,      length  for  the  measurement  of  lines;  a  known 
A  square,    square  for  the  measurement  of  surfaces ;  and  a 
A  cube,     known  cube  for  the  measurement  of  solids.     The 
contents:   measure  or  contents  of  any  magnitude,  belong- 
how  ascer-    ing  to  either  class,  is  ascertained  by  finding  how 
many  times  that  magnitude  contains  its  unit  of 
measure. 

§  250.   There  is  yet  another  class  of  magni 
tudes  with  which  Geometry  is  conversant,  called 
Angles:     Angles.      They   are    not,    however,   elementary 
magnitudes,  but  arise  from  the  relative  positions 
Their  unit,   of   those    already  described.     The  unit  of  this 
class  is  the  right  angle  ;  and  with  this  as  a  stand 
ard,  all  other  angles  are  compared 

§  251.  We  have  dwelt  with  much  detail  on 

the  unit  of  measure,  because   it   furnishes   the 

importance  only  basis  of   estimating    quantity.      The  con- 

of  the  unit  of 

measure:  ception  of  number  and  space  merely  opens  to 
the  intellectual  vision  an  unmeasured  field  of 
investigation  and  thought,  as  the  ascent  to  the 
summit  of  a  mountain  presents  to  the  eye  a 


CHAP.    III.]  GEOMETRY.  231 

wide  and  unsurveyed  horizon.     To  ascertain  the  space 

i      •    i          r-     i  .  f       •  IT  r     i         n*te 

height  01  the  point  01  view,  the  diameter  ol  the  i 
surrounding  circular  area  and  the  distance  to 
any  point  which  may  be  seen,  some  standard  or 
unity  must  be  known,  and  its  value  distinctly 
apprehended.  So,  also,  number  and  space,  which 
at  first  fill  the  mind  with  vague  and  indefinite 

measured 

conceptions,  are  to  be  finally  measured  by  units       by  u. 
of  ascertained  value. 


§  252.    It  is  found,   on  careful  analysis,  that  Every  num- 
every  number  may  be  referred  to  the  unit  one,  * 


as  a  standard,  and  when  the  signification  of  the  the  unit  one- 
term  ONE  is  clearly  apprehended,  that  any  num 
ber,  whether  large  or  small,  whether  integer  or 
fractional,  may  be  deduced  from  the  standard  by 
an  easy  and  known  process. 

In  space,   also,  which  is  indefinite  in  extent,      Space: 
and  exactly  similar  in  all  its  parts,  the  faculties 
of  the  mind  have  established  ideal  boundaries,     its  ideal 
These  boundaries  give  rise  to  the  geometrical 
magnitudes,  each  of  which  has  its  own  unit  of 
measure  ;  and  by  these  simple  contrivances,  we 
measure  space,  even  to  the  stars,  as  with  a  yard 
stick. 

§  253.  We  have,  thus  far,  not  alluded  to  the 
difficulty  of  determining  the  exact  length  of  that 


232  MATHEMATICAL     SCIENCE.  [fiOOK   II. 


Conception    which  we  regard  as  a  standard.     We  are  pre- 

o!  the  unit  of 

measure:  sented  with  a  given  length,  and  told  that  it  is 
called  a  foot  or  a  yard,  and  this  being  usually 
done  at  a  period  of  life  when  the  mind  is  satis 
fied  with  mere  facts,  we  adopt  the  conception 

At  first,  a     of  a  distance  corresponding  to  a  name,  and  then 

^"sion!  9"  ^  multiplying  and  dividing  that  distance  we 
are  enabled  to  apprehend  other  distances.  But 
this  by  no  means  answers  the  inquiry,  What  is 
the  standard  for  measurement  ? 

HOW  deter-  Under  the  supposition  that  the  laws  of  phys- 
ied*  ical  nature  operate  uniformly,  the  unit  of  meas 
ure  in  England  and  the  United  States  has  been 
fixed  by  ascertaining  the  length  of  a  pendulum 
which  will  vibrate  seconds,  and  to  this  length 
the  Imperial  yard,  which  we  have  also  adopted 
as  a  standard,  is  referred.  Hence,  the  unit  of 
•  What  it  is.  nieasure  is  referred  to  a  natural  standard,  viz.  to 
the  distance  between  the  axis  of  suspension  and 
the  centre  of  oscillation  of  a  pendulum  which 
shall  vibrate  seconds  in  vacuo,  in  London,  at  the 
level  of  the  sea.  This  distance  is  declared  to 

its  length,  be  39.1303  imperial  inches;  that  is,  3  imperial 
feet  and  3.1393  inches.  Thus,  the  determina- 

Difficuities    lion  of  the  unit  of  length  demands  the  applica- 
bwtt.      li°n  °f  lne  mosl  abstruse  science,  combined  with 
accurate  observation  and  delicate  experiment. 
Could  this  distance,  or  unit,  have  been  exactly 


CHAP.  III.]  GEOMETRY.  233 


ascertained  before  the   measures   of   the  world 

were  fixed,  and  in  general  use,   it  would  have  what  should 

„  .  have  been 

anorded  a  standard  at  once  certain  and  conve- 


nient,  and  all  distances  would  then  have  been  other  num- 

.  .    .  ,,  .  .  hers  derived 

expressed  in  numbers  arising  from  its  multiph-      fromit. 

cation  or  exact  division.     But  as  the  measures 

of  the  world  (and  consequently  their  units)  were  why  it  is  not 

fixed  antecedently  to  the  determination  of  this 

distance,  it  was  expressed  in  measures  already 

known ;  and  hence,  instead  of  being  represented 

by   1,  which  had  already  been  appropriated  to   what  now 

represents  it. 

the  foot,  it  was  expressed  in  terms  of  the  foot, 
viz.  39.1393  inches,  and  this  is  now  the  standard 
to  which  all  units  of  measure  are  referred. 

§  254.  The  unit  of  measure  is  not  only  im-  unit  of  meas 
ure  the  basis 

portant  as  affording  a  basis  for  all  measurement,  of  the  unit  of 
but  is  also  the  element  from  which  we  deduce 
the  unit  of  weight.    The  weight  of  27.7015  cubic 
inches  of  distilled  water  is  taken  as  the  standard, 
weighing  exactly  one  pound  avoirdupois,  and  this 
quantity  of  water  is  determined  from  the  unit 
of  length ;  that  is,  the  determination  of  it  reaches   what  it  is. 
back  to  the  length  of  a  pendulum  which  will 
vibrate  seconds  in  the  latitude  of  London. 

§  255.  Two  geometrical  figures  are  said  to  be   Equivalent 
equivalent,  when  they  contain  the  same  unit  of 


234  MATHEMATICAL     SCIENCE.  [BOOK   II. 

measure  an  equal  number  of  times.  Two  figures 
Equal  fig-  are  said  to  be  equal  when  they  can  be  so  applied 

to  each  other  as  to  coincide  throughout    their 

Equivalency:  whole    extent.      Hence,   equivalency   refers    to 

Equality.     measure>  and  equality  to  coincidence.     Indeed, 

coincidence  is  the  only  test  of  geometrical  equal 

ity.  All  equal  figures  are  of  course  equivalent, 
Their  differ-  though  equivalent  figures  are  by  no  means  equal. 

Equality  is  equivalency,  with  the  further  mark 

of  coincidence. 

PROPERTIES     OF     FIGURES. 

Property  of        §  256.  A  property  of  a  figure  is  a  mark  cora- 

figures. 

mon  to  all  figures  of  the  same  class.     For  exam- 


pje  :  if  the  class  be  "  Quadrilateral,"  there  are  two 

als. 

very  obvious  properties,  common  to  all  quadri 
laterals,  besides  the  one  which  characterizes 
the  figure,  and  by  which  its  name  is  defined, 
viz.  that  it  has  four  angles,  and  that  it  may 
be  divided  into  two  triangles.  If  the  class  be 

Pwaiieio-  "  Parallelogram,"  there  are  several  properties 
common  to  all  parallelograms,  and  which  are 
subjects  of  proof;  such  as,  that  the  opposite 
sides  and  angles  are  equal  ;  the  diagonals  divide 
each  other  into  equal  parts,  &c.  If  the  class  be 

Triangle:  "Triangle,"  there  are  many  properties  common 
to  all  triangles,  besides  the  characteristic  that 


CHAP.  III.]  GEOMETRY.  235 

they  have  three  sides.     If  the  class  be  a  par-   Equilateral, 
ticular  kind  of  triangle,  such  as  the  equilateral,    isosceles, 
isosceles,  or  right-angled  triangle,  then  each  class  Right-angled. 
has  particular  properties,  common  to  every  indi 
vidual  of  the  class,  but  not  common  to  the  other 

classes.      It  is  important,  however,   to   remark,  Every  prop 
erty  which 
that  every  property  which  belongs  to  "  triangle,"  belongs  to  a 

,     ,  ...  .  genus  will  be 

regarded  as    a  genus,  will    appertain    to   every   common  to 
species    or   class   of  triangle ;    and   universally,    ev^gs.pe~ 
every  property  which  belongs   to  a  genus  will 
belong   to    every   species  under  it ;    and    every 
property  which    belongs  to   a   species  will    be 
long  to  every  class  or  subspecies  under  it;  and  aisoto  every 


every  property  which  belongs  to  one  of  a  sub- 
species  or-  class  will  be  common  to  every  indi-   individual- 
vidual  of  the  class.     For  example  :  "  the  square    Examples. 
on  the  hypothenuse  of  a  right-angled  triangle  is 
equivalent  to  the  sum  of  the  squares  described 
on  the  other  two  sides,"  is  a  proposition  equally 
true  of  every  right-angled  triangle :  and  "  every 
straight  line    perpendicular  to   a   chord,   at  the      circle, 
middle  point,  will  pass  through  the  centre,"  is 
equally  true  of  all  circles. 


MARKS  OF  WHAT  MAY  BE  PROVED. 

§  257.  The  characteristic  properties  of  every 
geometrical  figure  (that  is,  those  properties  with-   tlc  tPe°per 


236  MATHEMATICAL     SCIENCE.  [BOOK  II. 

out  which  the  figures  could  not  exist),  are  given 
in  the  definitions.     How  are  we  to  arrive  at  all 
the   other   properties    of    these    figures?      The 
propositions  implied  in  the  definitions,  viz.  that 
Marks:      things  corresponding  to  the  words  defined  do  or 
may  exist  with  the  properties  named  ;  and  the 
or  what  may  self-evident  propositions  or  axioms,  contain  the 

be  proved. 

only  marks  of  what  can  be  proved  ;  and  by   a 
HOW  ex-     gkjjfui  combination  of  these  marks  we  are  able 

tended. 

to  discover  and  prove  all  that  is  discovered  and 
proved  in  Geometry. 

Definitions  and  axioms,  and  propositions  de- 
premiss      duced  ^rom    them,   are  major  premises  in  each 


The  science:  new  demonstration;  and  the  science  is  made  up 
consists.  °f  the  processes  employed  for  bringing  unfore 
seen  cases  under  these  known  truths  ;  or,  in  syl 
logistic  language,  for  proving  the  minors  neces 
sary  to  complete  the  syllogisms.  The  marks 
being  so  few,  and  the  inductions  which  furnish 
them  so  obvious  and  familiar,  there  would  seem 
to  be  very  little  difficulty  in  the  deductive  pro 
cesses  which  follow.  The  connecting  together 
of  several  of  these  marks  constitutes  Deductions, 

Geometry,    or  Trains  of  Reasoning;  and  hence,  Geometry 

a  Deductive    •          TV     i       *•         ci    • 

science.     ls  a  Deductive  Science. 


CHAP.   III.]  GEOMETRY.  237 


DEMONSTRATION. 

§  258.  As  a  first  example,  let  us  take  the  first 
proposition  in  Legendre's  Geometry : 

"If  a  straight  line  meet  another  straight  line,  Proposition 
the  sum  of  the  two  adjacent  angles  will  be  equal 
to  two  right  angles." 

Let  the  straight  line  DC 

meet  the  straight  line  AB  Enunciation. 

at  the  point  C,  then  will  the 
angle  ACD  plus  the  angle 
DCB  be  equal  to  two  right  AC  B 

angles. 

To  prove  this  proposition,  we  need  the  defini-      Thing9 

necessary  to 

tion  of  a  right  angle,  viz. :  prove  it. 

When  a  straight  line  AB 
meets  another  straight  line 
CD,  so  as  to  make  the  ad 
jacent  angles  BAG  and 


BAD  equal  to  each  other, 

each  of  those  angles  is  called  a  RIGHT  ANGLE,  and 

the  line  AB  is  said  to  be  PERPENDICULAR  to  CD. 

We  shall  also  need  the  2d,  3d,  and  4th  axioms,     Axioms, 
for  inferring  equality,*  viz.  : 

2.  Things  which  are  equal  to  the  same  thing     second, 
are  equal  to  each  other. 

f  Section  102. 


238  MATHEMATICAL     SCIENCE.  [BOOK  II. 

Third.          3.    A  whole  is   equal   to  the  sum  of  all  its 

parts. 

Fourth.         4.    If  equals   be  added   to   equals,   the  sums 
will  be  equal. 

Now  before  these  formulas  or  tests  can  be  ap- 
Linetobe    plied,  it  is  necessary  to  sup-  E        D 

pose  a  straight  line  CE  to  be 
Proof:      drawn  perpendicular  to  AB 
at  the  point  C :  then  by  the 
definition  of  a   right   angle,      A  C          B 

the  angle  ACE  will  be  equal  to  the  angle  ECB. 

By  axiom  3rd,  we  have, 

continued:  ACD  equal  to  ACE  plus  ECD :  to  each  of 
these  equals  add  DCB ;  and  by  the  4th  axiom 
we  shall  have, 

ACD  plus  DCB  equal  to  ACE  plus  ECD  plus 
DCB  ;  but  by  axiom  3rd, 

ECD   plus  DCB   equals  ECB:    therefore  by 
axiom  2d, 

ACD  plus  DCB  equals  ACE  plus  ECB. 
But  by  the  definition  of  a  right  angle, 
conclusion.       ACE  plus  ECB  equals  two  right  angles  :  there 
fore,  by  the  2d  axiom, 

ACD  plus  DCB  equals  two  right  angles, 
its  bases.        It  will  be  seen  that  the  conclusiveness  of  the 

proof  results, 

First.  1st.  From  the  definition,  that  ACE  and  ECB 

are  equal   to  each  other,   and  each  is  called  a 


CHAP.  III.]  GEOMETRY.  239 

right-angle  :  consequently,  their  sum  is  equal  to 
two  right  angles  ;  and, 

2dly.  In  showing,  by  means  of  the  axioms,  that     second. 
ACD  plus  DCB  equals  ACE  plus  ECB;    and 
then  inferring  from  axiom  2d,  that,  ACD  plus 
DCB  equals  two  right  angles. 

§259.  The  difficulty  in  the  geometrical  rea-  Difficulties  in 

the  demon- 

soning  consists  mainly  in  showing  that  the  prop-     stations. 
osition  to  be  proved  contains  the  marks  which 
prove  it.     To   accomplish  this,   it  is  frequently 
necessary  to  draw  many  auxiliary  lines,  forming   Auxiliaries 
new  figures  and  angles,  which  can  be  shown  to 
possess  marks  of  these  marks,   and  which  thus 
become   connecting   links   between   the   known   connecting 
and  the  unknown  truths.     Indeed,  most  of  the 
skill  and  ingenuity  exhibited  in  the  geometrical 
processes  are  employed  in  the  use  of  these  auxil 
iary  means.    The  example  above  affords  an  illus 
tration.     We  were  unable  to  show  that  the  sum   HOW  used, 
of  the  two  angles  possessed  the  mark  of  being 
equal  to  two  right  angles,  until  we  had  drawn  a 
perpendicular,   or   supposed   one   drawn,  at   the 
point  where  the  given  lines  intersect.     That  be 
ing  done,  the  two  right  angles  ACE  and  ECB  conclusion, 
were  formed,  which  enabled  us  to  compare  the 
sum  of  the  angle  ACD  and  DCB  with  two  right 
angles,  and  thus  we  proved  the  proposition. 


240  MATHEMATICAL     SCIENCE.  [BOOK  II. 

Proposition.       §  260.  As  a  second  example,  let  us  take  the 
following  proposition : 

Enunciation.       If  two  straight  lines  meet  each  other,  the  op 
posite  or  vertical  angles  will  be  equal. 

Let   the    straight  line 
AB  meet  the  straight  line 

Diagram.       j?T\  '  r1         U 

will   the  angle    ACD  be 

JE 

equal  to  the  opposite  an 
gle  ECB ;  and  the  angle  ACE  equal  to  the  an 
gle  DCB. 

Principles        To  prove  this  proposition,  we  need  the  last 
isary'    proposition,  and  also  the  2d  and  5th  axioms,  viz. : 

"  If  a  straight  line  meet  another  straight  line, 
the  sum  of  the  two  adjacent  angles  will  be  equal 
to  two  right  angles." 

Axioms.         «  Things  which  are  equal  to  the  same  thing 
are  equal  to  each  other." 

"  If  equals  be  taken  from  equals,  the  remain 
ders  will  be  equal." 

Now,  since  the   straight   line    AC   meets  the 
straight  line  ED  at  the  point  C,  we  have, 
proof.          ACD  plus  ACE  equal  to  two  right  angles. 

And  since  the  straight  line  DC  meets  the 
straight  line  AB,  we  have, 

ACD  plus  DCB  equal  to  two  right  angles : 
hence,  by  the  second  axiom, 

ACD  plus  ACE  equals  ACD  plus  DCB  :  ta- 


CHAP.  III.]  GEOMETRY.  211 

king  from    each  the  common   angle    ACD,  we  conclusion, 
know    from   the    fifth   axiom    that  the  remain 
ders   will   be   equal ;    that   is,    the   angle    ACE 
equal  to  the  opposite  or  vertical  angle  DCB. 

§  261.    The  two  demonstrations  given  above 
combine  all  the  processes  of  proof  employed  in  Demonstra- 
every  demonstration  of  the  same  class.     When  tions  generaL 
any  new  truth  is  to  be  proved,  the  known  tests 
of    truth    are   gradually   extended   to    auxiliary  Useof  auxil. 
quantities   having  a   more  intimate    connection   iaryquar 
with  such  new  truth  than  existed  between  it  and 
the  known  tests,  until  finally,  the  known  tests, 
through  a  series  of  links,  become  applicable  to 
the  final  truth  to  be  established :    the  interme 
diate   processes,    as   it  were,  bridging  over  the 
space  between  the   known   tests    and  the  final 
truth  to  be  proved. 

§  262.    There  are  two  classes  of  demonstra-  Direct  dera 
tions,  quite  different  from  each  other,   in  some 
respects,   although   the  same  processes  of  argu 
mentation  are  employed  in  both,  and  although 
the   conclusions   in   both   are   subjected   to   the 
same  logical  tests.     They  are  called  Direct,  or    Ne  ^^ 
Positive   Demonstration,   and  Negative  Demon-        ™ 

Reductio  ad 

stration,  or  the  Reductio  ad  Absurdum. 


16 


242  MATHEMATICAL     SCIENCE.  [BOOK   II. 


Difference.        §   263.     The    main    differences   in   the    two 
methods  are  these  :  The  method  of  direct  demon- 
Direct  Dem-  stration  rests  its  arguments  on  known  and  ad- 

onstration. 

mitted    truths,  and  shows   by  logical    processes 

that  the  proposition  can  be  brought  under  some 

previous  definition,  axiom,  or  proposition  :  while 

Negative     the  negative  demonstration  rests  its  arguments 

Demonstra 
tion,       on  an  hypothesis,  combines  this  with  known  pro 
positions,  and  deduces  a  conclusion  by  processes 

conclusion:  strictly  logical.  Now  if  the  conclusion  so  de 
duced  agrees  with  any  known  truth,  we  infer 

^"hjehdat  that  the  hypothesis,  (which  was  the  only  link  in 
the  chain  not  previously  known),  was  true ;  but 
if  the  conclusion  be  excluded  from  the  truths 
previously  established ;  that  is,  if  it  be  opposed 
to  any  one  of  them,  then  it  follows  that  the  hy 
pothesis,  being  contradictory  to  such  truth,  must 

Determines   ^Q  fa}S6i     jn  the  negative  demonstration,  there- 

whether  the 

hypothesis  is  fore,  the  conclusion  is  compared  with  the  truths 

true  or  false. 

known  antecedently  to  the  proposition  in  ques 
tion  :  if  it  agrees  with  any  one  of  them,  the  hy 
pothesis  is  correct ;  if  it  disagrees  with  any  one 
of  them,  the  hypothesis  is  false. 


proof  by         §  264.  We  will  give  for  an  illustration  of  this 

Negative 

emonstra-  method,  Proposition  XVII.  of  the  First  Book  of 
Legendre :  "  When  two  right-angled  triangles 
have  the  hypothenuse  and  a  side  of  the  one  equal 


CHAP.  III.]  GEOMETRY.  243 

to  the  hypothenuse  and  a  side  of  the  other,  each  Enunciation, 
to  each,  the  remaining  parts  will  be  equal,  each  to 
each,  and  the  triangles  themselves  will  be  equal." 

In  the  two   right-angled  triangles  BAG   and 
EDF  (see  next  figure),  let  the  hypothenuse  AC  Enunciation 
be  equal  to  DF,  the  side  BA  to  the  side  ED:  bythefigure' 
then  will  the  side  BC  be  equal  to  EF,  the  angle 
A  to  the  angle  D,  and  the  angle  C  to  the  angle  F. 
To  prove  this  proposition,  we  need  the  follow 
ing,  which  have  been  before  proved  ;  viz. : 

Prop.  X.  (of  Legendre).   "When  two  triangles     Previous 
have  the  three  sides  of  the  one  equal  to  the  three  trutbs  neces" 

sary. 

sides  of  the  other,  each  to  each,  the  three  an 
gles  will  also  be  equal,  each  to  each,  and  the 
triangles  themselves  will  be  equal." 

Prop.    V.     "  When   two    triangles   have   two  Proposition 
sides  and  the  included  angle  of  the  one,  equal 
to  two  sides  and  the  included  angle  of  the  other, 
each  to  each,  the  two  triangles  will  be  equal." 

Axiom  I.    "  Things  which  are   equal   to   the     Axioms, 
same  thing,  are  equal  to  each  other." 

Axiom   X.   (of  Legendre).    "All  right  angles 
are  equal  to  each  other." 

Prop.  XV.  "  If  from  a  point  without  a  straight  Proposition, 
line,  a  perpendicular  be  let  fall  on  the  line,  and 
oblique  lines  be  drawn  to  different  points, 

1st.  "  The  perpendicular  will  be  shorter  than 
any  oblique  line  ; 


244  MATHEMATICAL     SCIENCE.  [BOOK  II. 

2d.  "  Of  two  oblique  lines,  drawn  at  pleasure, 
that  which  is  farther  from  the  perpendicular  will 
be  the  longer." 

Now  the  two  sides  BC  and 
Beginning  of  EF  are  either  equal  or  un- 

the  demon 
stration,      equal.       If   they   are  equal, 

then  by  Prop.  X.  the  remain 
ing  parts  of  the  two  trian-  c  G  B  F  E 
gles  are  also  equal,  and  the  triangles  themselves 
are  equal.  If  the  two  sides  are  unequal,  one  of 
them  must  be  greater  than  the  other:  suppose 
BC  to  be  the  greater. 

construction  On  the  greater  side  BC  take  a  part  BG,  equal 
e'  to  EF,  and  draw  AG.  Then,  in  the  two  trian 
gles  BAG  and  DEF  the  angle  B  is  equal  to  the 
angle  E,  by  axiom  X  (Legendre),  both  being 
right  angles.  The  side  AB  is  equal  to  the  side 
DE,  and  by  hypothesis  the  side  BG  is  equal  to  the 
side  EF :  then  it  follows  from  Prop.  V.  that  the 
side  AG  is  equal  to  the  side  DF.  But  the  side 

DeuonStra"  DF  is  e(lual  to  the  side  AC  :  hence'  by  axiom  *> 
the  side  AG  is  equal  to  AC.     But  the  line  AG 

cannot  be  equal  to  the  line  AC,  having  been 
shown  to  be  less  than  it  by  Prop.  XV. :  hence, 
conclusion,  the  conclusion  contradicts  a  known  truth,  and  is 
therefore  false ;  consequently,  the  supposition  (on 
which  the  conclusion  rests),  that  BC  and  EF  are 
unequal,  is  also  false  ;  therefore,  they  are  equal. 


CHAP.   III.]  GEOMETRY.  245 

§  265.  It  is  often  supposed,  though  erroneous-     Negative 
ly,  that  the  Negative  Demonstration,  or  the  dem-       tion\ 
onstration  involving  the  "  reductio  ad  absurdum," 
is  less  conclusive  and  satisfactory  than  direct  or   conclusive, 
positive  demonstration.     This  impression  is  sim 
ply  the  result  of  a  want  of  proper  analysis.     For 
example  :  in  the  demonstration  just  given,  it  was     Reasons, 
proved  that  the  two  sides   BC  and  EF  cannot 
be  unequal;  for,  such  a  supposition,  in  a  logi 
cal  argumentation,  resulted  in  a  conclusion  di-    conclusion 

corresponds 

redly  opposed  to  a  known  truth ;  and  as  equality  to?  or  is  op_ 
and  inequality  are  the  only  general  conditions     p°sedto 

known  truth. 

of  relation  between  two  quantities,  it  follows 
that  if  they  do  not  fulfil  the  one,  they  must  the 
other.  In  both  kinds  of  demonstration,  the 
premises  and  conclusion  agree  ;  that  is,  they  are  Agreement, 
both  true,  or  both  false ;  and  the  reasoning  or 
argument  in  both  is  supposed  to  be  strictly  logi 
cal. 

In  the  direct  demonstration,  the  premises  are 
known,    being   antecedent    truths ;    and    hence, 
the  conclusion  is  true.     In  the  negative  demon-  Differences  in 
stration,  one  element   is    assumed,  and  the  con-       keirj° 
elusion  is  then  compared  with  truths  previously 
established.     If  the  conclusion  is  found  to  agree 
with  any  one   of  these,  we  infer  that   the  hy-    when  the 
pothesis  or  assumed  element  is  true ;  if  it  con- 
tradicts  any  one  of  these  truths,  we  infer  that 


246 


MATHEMATICAL     SCIENCE.  [fiOOK  II. 


tion. 


when  false,  the  assumed  element  is  false,  and  hence  that  its 
opposite  is  true. 

Measured:  §  266.  Having  explained  the  meaning  of  the 
term  measured,  as  applied  to  a  geometrical  mag 
nitude,  viz.  that  it  implies  the  comparison  of  a 
magnitude  with  its  unit  of  measure ;  and  having 
also  explained  the  signification  of  the  word  Prop 
erty,  and  the  processes  of  reasoning  by  which, 
in  all  figures,  properties  not  before  noticed  are 
inferred  from  those  that  are  known ;  we  shall 
now  add  a  few  remarks  on  the  relations  of  the 
geometrical  figures,  and  the  methods  of  compar 
ing  them  with  each  other. 


General 
Remarks. 


PROPORTION     OF     FIGURES. 

Proportion:        §  267.    Proportion  is   the  relation  which  one 

geometrical  magnitude  bears  to  another  of  the 

same  kind,  with  respect  to  its  being  greater  or 

less.    The  two  magnitudes  so  compared  are  called 

its  measure,  terms,  and  the  measure  of  the  proportion  is  the 

quotient  which  arises  from  dividing  the  second 

Ratio       term  by  the  first,  and  is  called  their  Ratio.     Only 

Quantities  of  quantities  °f  the  same  kind  can   be  compared 

the  same     together,  and  it  follows  from  the  nature  of  the 

kind  com 
pared,       relation   that  the  quotient  or  ratio  of  any  two 

terms  will  be  an  abstract  number,  whether  the 
terms  themselves  be  abstract  or  concrete. 


CHAP.   III.]  GEOMETRY.  217 

§  268.  The  term  Proportion  is  defined  by  most  Proportion: 

, .  ,,  .          i  c  how  defined. 

authors,  "An  equality  of  ratios  between  lour 
numbers  or  quantities,  compared  together  two 
and  two."  A  proportion  certainly  arises  from 
such  a  comparison  :  thus,  if 

_  =  __;    then,  Example. 

A       U 
A  :  B  :  :  C  :  D 

is  a  proportion. 

But  if  we  have  two  quantities  A  and  B,  which   True  defim- 

lion. 

may  change  their  values,  and  are,  at  the  same 
time,  so  connected  together  that  one  of  them 
shall  increase  or  decrease  just  as  many  times  as 
the  other,  their  ratio  will  not  be  altered  by  such 
changes ;  and  the  two  quantities  are  then  said  JJJf'iS. 
to  be  in  proportion,  or  proportional. 

Thus,  if  A  be  increased  three  times  and  B 
three  times,  then, 

3  B_A 
3A~B; 

that  is,  3  A  and  3  B  bear  to  each  other  the  same 
proportion  as  A  and  B.  Science  needed  a  gen-  Term  need- 

till 

eral  term  to  express  this  relation  between  two 
quantities  which  change   their   values,  without 
altering  their  quotient,   and  the  term  "propor 
tional,"  or  "  in  proportion,"  is  employed  for  that   How  used- 
purpose. 


248  MATHEMATICAL     SCIENCE.  [BOOK   II. 

Reasons  for       As  some  apology  for  the  modification  of  the 

'  definition  of  proportion,  which  has  been  so  long 

accepted,  it  may  be  proper  to  state  that  the  term 

has  been  used  by  the  best  authors  in  the  exact 

use  of  the   sense  here  attributed  to  it.     In  the  definition  of 

term. 

the  second  law  of  motion,  we  have,  "  Motion, 
or  change  of  motion,  is  proportional  to  the  force 
impressed  ;"*  and  again,  "  The  inertia  of  a  body 
is  proportioned  to  its  weight."f  Similar  exam 
ples  may  be  multiplied  to  any  extent.  Indeed, 
symbol  used  there  is  a  symbol  or  character  to  express  the 

to  represent 

proportion,  relation  between  two  quantities,  when  they  un 
dergo  changes  of  value,  without  altering  their 
ratio.  That  character  is  oc,  and  is  read  "  pro 
portional  to."  Thus,  if  we  have  two  quantities 
denoted  by  A  and  B,  written, 

Example.  A  OC  B, 

the  expression  is  read,  "  A  proportional  to  B." 
Another  kind       §  269.  There  is  yet  another  kind  of  relation 

of  proper-          1-1  •        i 

tion.  which  may  exist  between  two  quantities  A  and 
B,  which  it  is  very  important  to  consider  and 
understand.  Suppose  the  quantities  to  be  so 
connected  with  each  other,  that  when  the  first 
is  increased  according  to  any  law  of  change,  the 
second  shall  decrease  according  to  the  same  law ; 
and  the  reverse. 

*  Olmsted's  Mechanics,  p.  28.  f  Ibid.  p.  23. 


CHAP.  III.] 


GEOMETRY. 


249 


For  example  :  the  area  of  a  rect 
angle  is  equal  to  the  product  of  its 
base  and    altitude.     Then,    in    the 
rectangle  ABCD,  we  have 

Area  =  AB  x  BC. 

Take  a    second  rectangle   EFGH,  having  a     gecond 
longer  base  EF,  and  a  less  altitude  FG,  but  such    ExamPle- 
that  it  shall   have   an   equal 
area  with  the  first :  then  we 
shall  have 

Area  =  EF  x  FG. 
Now  since  the  areas  are  equal,  we  shall  have 

AB  X  BC  =  EF  X  FG  ;  Equation. 

and  by  resolving  the  terms  of  this  equation  into 
a  proportion,  we  shall  have 

AB  :  EF  :  :  FG  :  BC.  Proportion. 

It  is  plain  that  the  sides  of  the  rectangle  ABCD 
may  be  so  changed  in  value  as  to  become  the 
sides  of  the  rectangle  EFGH,  and  that  while 
they  are  undergoing  this  change,  AB  will  in 
crease  and  BC  diminish.  The  change  in  the  Relations  of 

the  Quanti* 

values  of  these  quantities  will  therefore  take  place       ties.- 
according  to  a  fixed  law :  that  is,  one  will  be  di 
minished  as  many  times  as  the  other  is  increased, 


250  MATHEMATICAL     SCIENCE.  [BOOK  II. 

since  their  product  is  constantly  equal  to  the 
area  of  the  rectangle  EFGH. 
Expressed  by      Denote  the  side  AB  by  x  and  BC  by  y,  and 

lettera.  J  J   y 

the  area  of  the  rectangle  EFGH,  which  is  known, 
by  a;  then 

xy  =  a; 

and  when  the  product  of  two  varying  quantities 

is  constantly  equal  to  a  known  quantity,  the  two 

Reciprocal    quantities  are  said  to  be  Reciprocally  or  Inverse- 

inverse  Pro-  ty  proportional  ;  thus  x  and  y  are  said  to  be  in- 

Ion*     versely  proportional  to  each  other.     If  we  divide 

1   by  each  member  of  the  above  equation,  we 

shall  have 

J__l 

xy      a  ' 

Redactions   and  by  multiplying  both  members  by  x,  we  shall 

of  the 
Equations,      have 

1        X 


and  then  by  dividing  both  numbers  by  x,  we  have 


Final  form. 


that  is,  the  ratio  of  x  to  -  is  constantly  equal  to  -; 
that  is,  equal  to  the  same  quantity,  however  x  or 


CHAP.   III.]  GEOMETRY.  251 

y  may  vary ;  for,  a  and  consequently  -  does  not 
change.  Hence, 

Two  quantities,  which  may  change  their  values,      Inverse 
are  reciprocally  or  inversely  proportional,  when    Proportion 
one  is  proportional  to  unity  divided  by  the  other, 
and  then  their  product  remains  constant. 

We  express  this  reciprocal  or  inverse  relation 
thus: 

Aocl. 

A  is  said  to  be  inversely  proportional  to  B  :  the 
symbols  also  express  that  A  is  directly  propor 
tional  to  -Q.     If  we  have 
o 

— 

we  say,  that  A  is  directly  proportional  to  B,  and 

f^  Generally, 

inversely  proportional  to  G.  how  rea(L 

The  terms  Direct,  Inverse  or  Reciprocal,  ap 
ply  to  the  nature  of  the  proportion,  and  not  to 
the  Ratio,  which  is  always  a  mere  quotient  and 
the  measure  of  proportion.     The  term  Direct  ap-   Direct  and 
plies  to  all  proportions  in  which  the  terms  in-    te°nsno't 
crease  or  decrease  together ;  and  the  term  In-  applicable  to 

Ratio. 

verse  or  Reciprocal  to  those  in  which  one  term 
increases  as  the  other  decreases.  They  cannot, 
therefore,  properly  be  applied  to  ratio  without 
changing  entirely  its  signification  and  definition. 


252  MATHEMATICAL     SCIENCE.  [fiOOK  II. 


COMPARISON     OF     FIGURES. 

Geometrical       §  270.  In  comparing  geometrical  magnitudes, 

magnitudes 

compared,  by  means  of  their  quotient,  it  is  not  the  quotient 
alone  which  we  consider.  The  comparison  im 
plies  a  general  relation  of  the  magnitudes,  which 
is  measured  by  the  Ratio.  For  example :  we 

Example,  say  that  "  Similar  triangles  are  to  each  other  as 
the  squares  of  their  homologous  sides."  What, 
do  we  mean  by  that  ?  Just  this  : 

Formula  of       That  the  area  of  a  triangle 

Comparison.          T  _ 

Is  to  the  area  01  a  similar  triangle 

As  the  area  of  a  square  described  on  a  side  of 

the  first, 

To  the  area  of  a  square  described  on  an  ho 
mologous  side  of  the  second. 
Thus,  we  see  that  every  term  of  such  a  pro- 
changes  of  portion  is  in  fact  a  surface,  and  that  the  area 
how  affected  °f  a  triangle  increases  or  decreases  much  faster 
than  its  sides  ;  that  is,  if  we  double  each  side  of 
a  triangle,  the  area  will  be  four  times  as  great: 
if  we  multiply  each  side  by  three,  the  area  will 
Results,     be  nine    times  as  great ;    or  if  we  divide  each 
side  by  two,  we  diminish  the  area  four  times,  and 
so  on.     Again, 
circles  com-       The  area  of  one  circle 

Is  to  the  area  of  another  circle, 

As  a  square  described  on  the  diameter  of  the  first 


CHAP.  III.] 


GEOMETRY. 


253 


To  a  square  described  on  the  diameter  of  the 

second. 
Hence,  if  we  double  the  diameter  of  a  circle,    How  their 

areas  change. 

the  area  of  the  circle  whose  diameter  is  so  in 
creased  will  be  four  times  as  great :  if  we  mul 
tiply  the  diameter  by  three,  the  area  will  be  nine 
times  as  great ;  and  similarly  if  we  divide  the 
diameter. 


§  271.  In  comparing  solids  together,  the  same  comparison 
general  principles  obtain.     Similar  solids  are  to 
each  other  as  the  cubes  described  on  their  ho- 

That  is, 


Formula. 


mologous  or  corresponding  sides. 
A  prism 

Is  to  a  similar  prism, 
As  a  cube  described  on  a  side  of  the  first, 
Is  to  a  cube  described  on  an  homologous  side 

of  the  second. 
Hence,  if  the  sides  of  a  prism  be  doubled,  the    How  the 

solidities 

solid  contents  will  be  increased  eight-fold.    Again, 

A  sphere 

Is  to  a  sphere, 

As  a  cube  described  on  the  diameter  of  the  first, 

Is  to  a  cube  described  on  a  diameter  of  the 
second. 

Hence,  if  the  diameter  of  a  sphere  be  doubled, 
its  solid  contents  will  be  increased  eight-fold ;  if    changes. 
the  diameter  be   multiplied   by  three,   the  solid 


change. 


Sphere : 


How  its 
solidity 


254 


MATHEMATICAL     SCIENCE.  [fiOOK  II. 


contents  will  be  increased  twenty-seven  fold  : 
if  the  diameter  be  multiplied  by  four,  the  solid 
contents  will  be  increased  sixty-four  fold ;  the 
solid  contents  increasing  as  the  cubes  of  the 
numbers  1,  2,  3,  4,  &c. 


Ratio : 


an  abstract 
number. 


§  272.  The  relation  or  ratio  of  two  magnitudes 
to  each  other,  may  be,  and  indeed  is,  expressed 
by  an  abstract  number.  This  number  has  a 
fixed  value  so  long  as  we  do  not  introduce  a 

ing  a  fixed 

value.      change  in   the   volumes    of  the    solids ;    but   if 
we  wish  to  express  their  ratio   under  the  sup 
position    that    their    volumes    may   change    ac 
cording  to  fixed  laws  (that  is,  so  that  the  solids 
HOW  varying  shall  continue  similar),  we  then  compare  them 

solids  are 

compared,  with  similar  figures  described  on  their  homol 
ogous  or  corresponding  sides;  or,  what  is  the 
same  thing,  take  into  account  the  corresponding 
changes  which  take  place  in  the  abstract  num 
bers  that  express  their  volumes. 


General 
outline. 


Geometry : 


RECAPITULATION. 

§  273.  We  have  now  completed  a  general 
outline  of  the  science  of  Geometry,  and  what 
has  been  said  may  be  recapitulated  under  the 
following  heads.  It  has  been  shown, 

1st.  That  Geometry  is  conversant  about  space, 


CHAP.  III.]  GEOMETRY.  255 

or  those    limited   portions   of  space  which   are    to  what  u 
called  Geometrical  Magnitudes. 

2d.  That  the  geometrical  magnitudes  embrace 
three  species  of  figures  : 

1st.  Lines — straight  and  curved ;  Lines. 

2d.  Surfaces — plane  and  curved ;  surfaces. 

3d.  Solids — bounded  either   by  plane  sur-      solids, 
faces  or  curved,  or  both ;  and, 

4th.  Angles,  arising  from  the  positions  of     Angles, 
lines    and   planes,    and    by   which   they   are 
bounded. 

3d.  That  the  science  of  Geometry  is  made  up     science: 
of  those  processes  by  means  of  which  all   the        up 
properties  of  these  magnitudes  are  examined  and 
developed,  and  that  the  results  arrived  at  con 
stitute  the  truths  of  Geometry. 

4th.  That  the  truths  of  Geometry  may  be  di-      Truths: 
vided  into  three  classes  :  three  classes. 

1st.  Those  implied  in  the  definitions,  viz.    First  class. 
that    things   exist   corresponding   to   certain 
words  defined ; 

2d.    Intuitive   or    self-evident    truths   em-     second, 
bodied  in  the  axioms ; 

3d.  Truths  deduced  (that  is,  inferred)  from      TtiML 
the  definitions  and  axioms,  called  Demonstra 
tive  Truths. 

5th.  That  the  examination  of  the  properties  of  Geometrical 
the  geometrical  magnitudes  has  reference, 


256 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


Comparison.  1st.  To  their  comparison  with  a  standard 

or  unit  of  measure  ; 

Properties.  2d.  To  the  discovery  of  properties  belong 

ing  to  an  individual  figure,  and  yet  common  to 
the  entire  class  to  which  such  figure  belongs ; 

Proportion.  3d.  To  the  comparison,  with  respect  to  mag 

nitude,  of  figures  of  the  same  species  with  each 
other  ;  viz.  lines  with  lines,  surfaces  with  sur 
faces,  and  solids  with  solids. 


SUGGESTIONS  FOR  THOSE  WHO  TEACH  GEOMETRY. 

suggestions.        i    Be  sure  tna^  vour  pupijs  have  a  clear  ap- 
First.       prehension  of  space,  and  of  the  notion  that  Ge 
ometry  is  conversant  about  space  only. 

2.  Be  sure  that  they  understand  the  significa 
tion  of  the  terms,  lines,  surfaces,  and  solids,  and 
that  these    names   indicate  certain   portions  of 
space  corresponding  to  them. 

3.  See  that  they  understand  the  distinction  be 
tween   a  straight  line  and  a  curve;  between  a 
plane  surface  and  a  curved  surface ;  between  a 
solid  bounded  by  planes  and  a  solid  bounded  by 
curved  surfaces. 

4.  Be  careful  to  have  them  note  the  charac- 
Fourtn.     teristics  of  the  different  species  of  plane  figures, 

such  as  triangles,  quadrilaterals,  pentagons,  hexa 
gons,  &c. ;  and  then  the  characteristic  of  each 


Third. 


CHAP.   III.] 


GEOMETRY. 


257 


class  or  subspecies,  so  that  the  name  shall  recall, 
at  once,  the  characteristic  properties  of  each 
figure. 

5.  Be  careful,  also,  to   have  them  note   the 
characteristic   differences   of    the    solids.       Let      Firth 
them  often  name  and   distinguish   those   which 

are  bounded  by  planes,  those  bounded  by  plane 
and  curved  surfaces,  and  those  bounded  by 
curved  surfaces  only.  Regarding  Solids  as  a 
genus,  let  them  give  the  species  and  subspecies 
into  which  the  solid  bodies  may  be  divided. 

6.  Having  thus  made  them  familiar  with  the 
things  which  are  the  subjects  of  the  reasoning,      sixth. 
explain  carefully  the  nature  of  the  definitions ; 

then  of  the  axioms,  the  grounds  of  our  belief  in 
them,  and  the  information  from  which  those 
self-evident  truths  are  inferred. 

7.  Then  explain  to  them,  that  the  definitions 

and  axioms  are  the  basis  of  all  geometrical  rea-     geventh 
soning :  that  every  proposition  must  be  deduced 
from  them,  and  that  they  afford  the  tests  of  all 
the  truths  which  the  reasonings  establish. 

8.  Let  every  figure,  used  in  a  demonstration, 

be  accurately  drawn,  by  the  pupil  himself,  on  a     Eighth, 
blackboard.      This  will  establish    a   connection 
between  the  eye  and  the  hand,  and  give,  at  the 
same  time,  a  clear  perception  of  the  figure  and  a 

distinct  apprehension  of  the  relations  of  its  parts. 

17 


258  MATHEMATICAL     SCIENCE.  [BOOK  II. 

9.  Let  the  pupil,  in  every  demonstration,  first 
Ninth,      enunciate,  in  general  terms,  that  is,  without  the 

aid  of  a  diagram,  or  any  reference  to  one,  the 
proposition  to  be  proved ;  and  then  state  the 
principles  previously  established,  which  are  to 
be  employed  in  making  out  the  proof. 

10.  When  in  the  course  of  a  demonstration, 
Tenth.      any  truth  is  inferred  from  its  connection  with 

one  before  known,  let  the  truth  so  referred  to  be 
fully  and  accurately  stated,  even  though  the 
number  of  the  proposition  in  which  it  is  proved, 
be  also  required.  This  is  deemed  important. 

11.  Let  the  pupil  be  made  to  understand  that 
Eleventh,    a  demonstration  is  but  a  series  of  logical  argu 
ments    arising    from    comparison,   and  that    the 
result  of  every  comparison,  in  respect  to  quan 
tity,    contains   the   mark   either   of  equality  or 
inequality. 

12.  Let    the   distinction   between   a   positive 
Twelfth,     and  negative  demonstration  be  fully  explained 

and  clearly  apprehended. 

13.  In  the  comparison  of  quantities  with  each 
Thirteenth,   other,  great  care  should  be  taken  to  impress  the 

fact  that  proportion  exists  only  between  quan 
tities  of  the  same  kind,  and  that  ratio  is  the 
measure  of  proportion. 

14.  Do  not  fail  to  give  much  importance  to 
Fourteenth,  the  kind  of  quantity  under  consideration.     Let 


CHAP    III,]  GEOMETRY.  259 

the  question  be  often  put,  What  kind  of  quantity  Fourteenth, 
are  you  considering  ?     Is  it  a  line,  a  surface,  or 
a  solid  ?     And  what  kind  of  a  line,  surface,  or 
solid  ? 

15.  In  all  cases  of  measurement,  the  unit  of 
measure   should   receive    special   attention.      If 

lines  are  measured,  or  compared  by  means  of  a  Fifteenth. 
common  unit,  see  that  the  pupil  perceives  that 
unit  clearly,  and  apprehends  distinctly  its  rela 
tions  to  the  lines  which  it  measures.  In  sur 
faces,  take  much  pains  to  mark  out  on  the 
blackboard  the  particular  square  which  forms 
the  unit  of  measure,  and  write  unit,  or  unit  of 
measure,  over  it.  So  in  the  measurement  of 
solidity,  let  the  unit  or  measuring  cube  be  ex 
hibited,  and  the  conception  of  its  size  clearly 
formed  in  the  mind ;  and  then  impress  the  im 
portant  fact,  that,  all  measurement  consists  in 
merely  comparing  a  unit  of  measure  with  the 
quantity  measured ;  and  that  the  number  which 
expresses  the  ratio  is  the  numerical  expression 
for  that  measure. 

16.  Be  careful  to  explain  the  difference  of  the 
terms  Equal  and  Equivalent,  and  never  permit  sixteenth, 
the  pupil  to  use  them  as  synonymous.     An  ac 
curate  use  of  words  leads  to  nice  discriminations 

of  thought. 


CHAP.  IV.] 


ANALY  SIS. 


261 


CHAPTER    IV. 


ANALYSIS — ALGEBBA — ANALYTICAL    GEOMETRY DIFFERENTIAL   AND   INTEGRAL 

CALCULUS. 


ANALYSIS. 

§  274.  ANALYSIS  is  a  general  term,  embra 
cing  that  entire  portion  of  mathematical  science 
in  which  the  quantities  considered  are  repre 
sented  by  letters  of  the  alphabet,  and  the  opera 
tions  to  be  performed  on  them  are  indicated  by 
signs. 


Analysis 
defined. 


§  275.  We  have  seen  that  all  numbers  must    Numbers 

must  be  of 

be  numbers  of  something;*  for,  there  is  no  such      things; 
thing  as  a  number  without  a  basis  :  that  is,  every 
number  must  be  based  on  the  abstract  unit  one, 
or  on  some  unit   denominated.      But   although 
numbers  must  be  numbers  of  something,  yet  they  tut  may  be 

of  many  kuid 

may  be  numbers  of  any  thing,  for  the  unit  may    Of  things, 
be  whatever  we  choose  to  make  it. 


*  Section  112. 


262  MATHEMATICAL     SCIENCE.  [BOOK  II. 


AII  quantity       §  276.    All  quantity  consists   of  parts,   which 

consists  of  1 

parts.       can   be   numbered   exactly   or   approximative!}^, 

and,  in  this  respect,  possesses  all  the  properties 

of  number.     Propositions,   therefore,  concerning 

numbers,  have  the  remarkable  peculiarity,   that 

Propositions  they   are  propositions  concerning  all  quantities 

in  regard  to 

number     whatever.     That  half  of  six  is  three,  is  equally 

apply  also  to  , 

quantity.  true>  whatever  the  word  six  may  represent, 
whether  six  abstract  units,  six  men,  or  six  tri 
angles.  Analysis  extends  the  generalization  still 
further.  A  number  represents,  or  stands  for,  that 
particular  number  of  things  of  the  same  kind, 

Algebraic    without  reference  to  the  nature  of  the  thing ; 

symbols 

more  gener-  but  an  analytical  symbol  does  more,  for  it  may 
stand  for  all  numbers,  or  for  all  quantities  which 
numbers  represent,  or  even  for  quantities  which 
cannot  be  exactly  expressed  numerically. 

Anything        As  soon  as  we  conceive  of  a  thing  we  may 

conceived  .  .        ,..,,. 

may  be  di-   conceive   it  divided  into  equal  parts,  and  may 

Vlded'      represent  either  or  all  of  those  parts  by  a  or  x, 

or  may,  if  we  please,  denote  the  thing  itself  by  a 

or  x,  without  any  reference  to  its  being  divided 

into  parts. 


Each  figure       §277.    In  Geometry,  each  geometrical  figure 
class.  a  stands  for  a  class ;  and  when  we  have  demon 
strated  a  property  of  a  figure,  that  property  is 
considered  as  proved  for  every  figure  of  the  class. 


CHAP.  IV.]  ANALYSIS.  263 

For  example :  when  we  prove  that  the  square  Example. 
described  on  the  hypothenuse  of  a  right-angled 
triangle  is  equivalent  to  the  sum  of  the  squares 
described  on  the  other  two  sides,  we  demonstrate 
the  fact  for  all  right-angled  triangles.  But  in 
analysis,  all  numbers,  all  lines,  all  surfaces,  all  in  analysis 

the  symbols 

solids,  may  be  denoted  by  a  single  symbol,  a  or  x.    stand  for 

Hence,   all  truths   inferred   by  means   of   these     ^^ 

symbols  are  true  of  all  things  whatever,  and  not 

like   those  of  number  and  geometry,  true    only 

of  particular  classes  of  things.     It  is,  therefore, 

not  surprising,  that   the  symbols  of  analysis  do 

not  excite  in  our  minds  the  ideas  of  particular   Hence,  the 

•  j     truths  int'er- 

thmgs.     1  he  mere  written  characters,  a,  b,  c,  a,  red  are  geu_ 
x,  y,  z,  serve  as  the  representatives  of  things  in 
general,  whether  abstract  or  concrete,  whether 
known  or  unknown,  whether  finite  or  infinite. 

§  278.  In  the  uses  which  we  make  of  these     symbols 

come  to  be 

symbols,  and  the  processes  of  reasoning  carried   regarded  as 
on  by  means  of  them,  the  mind  insensibly  comes 
to  regard  them  as  things,  and  not  as  mere  signs ; 
and  we  constantly  predicate  of  them  the  prop 
erties  of  things  in  general,  without  pausing  to 
inquire  what  kind  of  thing  is  implied.      Thus,     Example, 
we   define   an  equation  to  be   a  proposition   in    Theequa- 
which  equality  is   predicated    of   one  thing    as 
compared  with  another.     For  example  : 


264  MATHEMATICAL     SCIENCE.  [BOOK  II. 


a  -f  c  =  x, 
whataxioms  *s   an    equation,   because  x   is    declared   to   be 

necessary  to    eqU£J  tO  the  sum    Qf    a    an(J  c        Jn  the    Solution  of 

its  solution.        • 

equations,  we  employ  the   axioms,  "  If  equals  be 

added  to  equals,  the  sums  will  be  equal ;"  and, 

"  If  equals  be  taken  from  equals,  the  remainders 

They  express  will  be  equal."     Now,  these  axioms  do  not  ex- 

qualitiesof 

things,      press   qualities   01    language,   but    properties   01 
Hence,  in-    quantity.      Hence,   all  inferences  in  mathemat- 

ferences  re-  . 

late  to  things,  ical  science,  deduced  through  the  instrumentality 
of  symbols,  whether  Arithmetical,  Geometrical, 
or  Analytical,  must  be  regarded  as  concerning 
quantity,  and  not  symbols. 

Quantity         As  analytical  symbols  are  the  representatives 
tewM-  °f  quantity  in  general,  there  is  no  necessity  of 
enttothe    keeping  the  idea  of  quantity  continually  alive  in 
the  mind ;    and  the  processes  of  thought  may, 
without  danger,  be  allowed  to  rest  on  the  sym 
bols  themselves,  and  therefore,  become  to  that 
extent,  merely  mechanical.     But,  when  we  look 
The  reason-  back  and  see  on  what  the  reasoning  is  based,  and 
based  o/the  now  tne  processes  have  been  conducted,  we  shall 
supposition   fin(j  tnat  every  step  was  taken  on  the  supposition 

of  quantity. 

that  we  were  actually  dealing  with  things,  and 
not  symbols ;  and  that,  without  this  understand 
ing  of  the  language,  the  whole  system  is  without 
signification,  and  fails. 


CHAP.  IV.]  ALGEBRA.  265 

§  279.  There  are  three  principal  branches  of     Three 

,       .  branches : 

Analysis  : 

1st.    Algebra.  Algebra, 

2d.  Analytical  Geometry.  Analytical 

*  Geometry, 

3d.  Differential  and  Integral  Calculus.  calculus. 

ALGEBRA. 

§  280.   Algebra  is,  in  fact,   a  species  of  uni-     Algebra: 
versal  Arithmetic,  in  which  letters  and  signs  are     universal 
employed  to  abridge  and  generalize  all  processes 
involving  numbers.     It  is  divided  into  two  parts,   TWO  parts: 
corresponding  to  the  science  and  art  of  Arith 
metic  : 

1st.  That  which  has  for  its  object  the  investi-    First  part: 
gation  of  the  properties  of  numbers,  embracing 
all   the  processes   of  reasoning  by  which   new 
properties  are  inferred  from  known  ones  ;  and, 

2d.  The  solution  of  all  problems  or  questions  second  part, 
involving  the  determination  of  certain  numbers 
which  are  unknown,  from  their  connection  with 
certain  others  which  are  known  or  given. 

ANALYTICAL     GEOMETRY. 


§  281.    Analytical   Geometry    examines   the  Analytical 

,                      ~     ,  Geometry. 

properties,  measures,   and  relations  ol  the  geo 
metrical  magnitudes  by  means  of  the  analytical  its  nature. 


266  MATHEMATICAL     SCIENCE.  [BOCK   II. 


symbols.     This  branch  of  mathematical  science 
Descartes,     originated  with  the  illustrious  Descartes,  a  cele- 

the  original 

founder  of    brated  French  mathematician  of  the  17th  ccn- 

this  science.  ..  .     ,  .  .   .  c          • 

tury.     He  observed  that  the  positions  01  points, 

What  he 

observed,    the  direction  of  lines,  and  the  forms  of  surfaces, 

could  be  expressed  by  means  of  the  algebraic 

AH  position  symbols  ;  and  consequently,  that  every  change, 

expressed  by 

symbols,  either  in  the  position  or  extent  of  a  geometrical 
magnitude,  produced  a  corresponding  change  in 
certain  symbols,  by  which  such  magnitude  could 
be  represented.  As  soon  as  it  was  found  that. 
to  every  variety  of  position  in  points,  direction 
in  lines,  or  form  of  curves  or  surfaces,  there  cor 
responded  certain  analytical  expressions  (called 
their  Equations),  it  followed,  that  if  the  processes 
were  known  by  which  these  equations  could  be 
The  equation  examined,  the  relation  of  their  parts  determined, 

develops  the 

properties    and   the  laws    according   to   which   those    parts 


Vai7»  relative  to  one  another,  ascertained,  then 
the  corresponding  changes  in  the  geometrical 
magnitudes,  thus  represented,  could  be  imme 
diately  inferred. 

Hence,  it  follows  that  every  geometrical  ques- 
Power  over   tion  can  be  solved,  if  we  can  resolve  the  corre- 

the  magni-  i       i        •  -•  j  ,1 

tude  extend-  sponding  algebraic  equation  ;  and  the  power  over 
edbythe    ^Q  geometrical  magnitudes  was  extended  iust  in 

equation. 

proportion  as  the  properties  of  quantity  were 
brought  to  light  by  means  of  the  Calculus.  The 


CHAP.   IV.]  ANALYSIS.  267 

applications  of  this  Calculus  were  soon  made  to  TO  what  sub 
ject  applied. 

the  subjects  of  mechanics,  astronomy,  and  in 
deed,  in  a  greater  or  less  degree,  to  all  branches 
of  natural  philosophy;  so  that,  at  the  present  its  present 
time,  all  the  varieties  of  physical  phenomena, 
with  which  the  higher  branches  of  the  science 
are  conversant,  are  found  to  answer  to  varieties 
determinate  by  the  algebraic  analysis. 


§  282.  Two  classes  of  quantities,  and  conse-    Quantities 

which  enter 

quently  two  sets  of  symbols,  quite  distinct  from  into  the  Cai- 

culus. 

each  other,   enter  into  this  Calculus ;    the   one 
called  Constants,  which  preserve  a  fixed  or  given    constants. 
value  throughout  the  same  discussion  or  investi 
gation  ;    and  the  other  called  Variables,  which    variables, 
undergo  certain  changes  of  value,   the  laws  of 
which  are  indicated  by  the  algebraic  expressions 
or  equations  into  which  they  enter.     Hence, 

Analytical  Geometry  may  be  defined  as  that    Analytical 
branch   of  mathematical  science,  which  exam-     defined, 
ines,   discusses,    and   develops  the  properties  of 
geometrical  magnitudes  by  noting  the  changes 
that  take  place  in  the  algebraic  symbols  which 
represent  them,  the  laws  of  change  being  deter 
mined  by  an  algebraic  equation  or  formula. 


208  MATHEMATICAL     SCIENCE.  [BOOK  II. 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 


Quantities        £  283    jn  fafe  branch  of  mathematical  science, 

considered. 

as  in  Analytical  Geometry,  two  kinds  of  quan- 
vanabies,    ^ity  are  considered,  viz.  Variables  and  Constants  ; 

Constants. 

and  consequently,  two  distinct  sets  of  symbols 
The  science,  are  employed.     The  science  consists  of  a  series 
of  processes  which  note  the  changes  that  take 
place  in   the   value  of    the   Variables.      Those 
changes  of  value  take  place  according  to  fixed 
laws  established  by  algebraic  formulas,  and  are 
Marks,      indicated  by  certain  marks  drawn  from  the  va- 
riable   symbols,    called    Differential  Coefficients. 


By  these  marks  we  are  enabled  to  trace  out  with 
the  accuracy  of  exact  science  the  most  hidden 
properties  of  quantity,  as  well  as  the  most  gen 
eral  and  minute  laws  which  regulate  its  changes 
of  value. 


Analytical        §  284.    It  will   be   observed,  that  Analytical 

and  '    Geometry  and  the  Differential  and  Integral  Cal- 

caicuius :    cu]us  treat  of  quantity  regarded  under  the  same 

general  aspect,  viz.  as  subject  to  changes  or  va- 

Howthey    nations  in  magnitude  according  to  laws  indica- 

regard  quan 
tity :       ted  by  algebraical  formulas;   and  the  quantities, 

whether  variable  or  constant,  are,  in  both  cases, 

by  what     represented  by  the  same  algebraic  symbols,  viz. 

ted'  the  constants  by  the  first,  and  the  variables  by 


CHAP.  IV.]  ALGEBRA.  269 


the  final  letters  of  the  alphabet.     There  is,  how-    Difference; 
ever,   this   important   difference :    in  Analytical 
Geometry  all  the  results  are  inferred  from  the    in  what  it 

m  consists. 

relations  which  exist  between  the  quantities 
themselves,  while  in  the  Differential  and  Integral 
Calculus  they  are  deduced  by  considering  what 
may  be  indicated  by  marks  drawn  from  variable 
quantities,  under  certain  suppositions,  and  by 
marks  of  such  marks. 

§  285.  Algebra,  Analytical  Geometry,  the  Dif-   Analytical 

Science. 

ferential  and  Integral  Calculus,  extended  into  the 
Theory  of  Variations,  make  up  the  subject  of 
analytical  science,  of  which  Algebra  is  the  ele 
mentary  branch.     As  the  limits  of  this  work  do     its  parts, 
not  permit  us  to  discuss  the  subject  in  full,  we 
shall  confine  ourselves  to  Algebra,  pointing  out, 
occasionally,  a  few  of  the  more  obvious  connec-     HOW  far 
tions  between  it  and  the  two  other  branches. 


ALGEBRA. 

§  286.  On  an  analysis  of  the  subject  of  Alge-     Algebra. 
bra,  we  think  it  will  appear  that  the  subject  itself 
presents  no  serious  difficulties,  and  that  most  of  Difficulties. 
the  embarrassment  which  is  experienced  by  the 
pupil  in  gaining  a  knowledge  of  its  principles,  as   How  over- 
well  as  in  their  applications,  arises  from  not  at- 


270  MATHEMATICAL     SCIENCE.  [BOOK  II. 

Language,    tending  sufficiently  to  the  language  or  signs  of 

the  thoughts  which  are  combined  in  the  reason 

ings.     At  the  hazard,  therefore,  of  being  a  little 

diffuse,  I  shall  begin  with  the  very  elements  of 

the  algebraic  language,  and  explain,  with  much 

minuteness,  the  exact  signification  of  the  char- 

characters    acters  that  stand  for  the  quantities  which  are  the 

LrtqJnttty!  subjects  of  the  analysis  ;  and  also  of  those  signs 

signs.       which  indicate  the  several  operations  to  be  per 

formed  on  the  quantities. 

Quantities.        §  287.  The  quantities  which  are  the  subjects 

HOW  divided,  of  the   algebraic    analysis  may  be  divided  into 

two  classes  :  those  which  are  known  or  given, 

and  those  which  are  unknown  or  sought.     The 

Howrepre-  known  are  uniformly  represented  by  the  first 
letters  of  the  alphabet,  a,  b,  c,  d,  &c.  ;  and  the 
unknown  by  the  final  letters,  x,  y,  z,  v,  w,  &c. 


May  be  in-        §  288'    Quantity  is   susceptible  of  being  in 
creased  or    creasec[  or  diminished  ;*  and  there  are  five  oper- 

diminished. 

Five  opera-  ations  which  can  be  performed  upon  a  quantity 
tions:       that  will  give  results  differing  from  the  quantity 

itself,  viz.  : 
First  1st.  To  add  it  to  itself  or  to  some  other  quan 

tity; 


*  Section  75. 


CHAP.   IV.] 


ALGEBRA. 


271 


2d.  To  subtract  some  other  quantity  from  it ;  second. 

3d.  To  multiply  it  by  a  number  ;  Third. 

4th.  To  divide  it ;  Fourth. 

5th.  To  extract  a  root  of  it.  Fifth- 

The  cases  in  which  the  multiplier  or  divisor 

is  1,  are  of  course  excepted  ;  as  also  the  case  Exception, 
where  a  root  is  to  be  extracted  of  1. 


§  289.  The  five  signs  which  denote  these  oper-       signs, 
ations  are  too  well  known  to  be  repeated  here. 
These,  with  the  signs  of  equality  and  inequality,    Elements 

of  the 

the  letters  of  the  alphabet  and  the  figures  which    Algebraic 
are  employed,  make  up  the  elements  of  the  alge 
braic  language.     The  words  and  phrases  of  the     its  words 

and  phrases: 

algebraic,  like  those  ot  every  other  language,  are 
to  be  taken  in  connection  with  each  other,  and 

are  not  to  be  interpreted  as  separate  and  isolated   HOW  inter 
preted, 
symbols. 


5»  290.  The  symbols  of  quantity  are  designed   symbols  of 
to  represent  quantity  in  general,  whether  abstract 
or  concrete,  whether  known  or  unknown;  and 
the  signs  which  indicate  the  operations  to   be     General. 
performed  on  the  quantities  are  to  be  interpreted 
in  a  sense  equally  general.     When  the  sign  plus 
is  written,  it  indicates  that  the  quantity  before   Examples, 
which  it  is  placed  is  to  be  added  to  some  other    signs  plus 
quantity ;  and  the  sign  minus  implies  the  exist- 


aud  minus. 


272  MATHEMATICAL     SCIENCE.  [fiOOK   II. 

ence  of  a  minuend,  from  which  the  subtrahend 

is  to  be  taken.     One  thing  should  be  observed  in 

signs  have   regard  to  the  signs  which  indicate  the  operations 

no  effect  on 

the  nature  of  that  are  to  be  performed  on  quantities,  viz.  they 

a  quantity. 

do  not  at  all  affect  or  change  the  nature  of  the 
quantity  before  or  after  which  they  are  written, 
but  merely  indicate  what  is  to  be  done  with  the 

Examples:  quantity.  In  Algebra,  for  example,  the  minus 
sign  merely  indicates  that  the  quantity  before 
which  it  is  written  is  to  be  subtracted  from 

in  Analytical  some  other  quantity ;  and  in  Analytical  Geom- 

Geometry. 

etry,  that  the  line  before  which  it  falls  is  esti 
mated  in  a  contrary  direction  from  that  in  which 
it  would  have  been  reckoned,  had  it  had  the  sign 

o 

plus  ;  but  in  neither  case  is  the  nature  of  the 
quantity  itself  different  from  what  it  would  have 
been  had  it  had  the  sign  plus, 
interpreta-        The  interpretation  of  the  language  of  Algebra 

tion  of  the  .        „  ,  .  .  .    .       , 

language:  1S  tne  first  thing  to  which  the  attention  of  a  pupil 
should  be  directed ;  and  he  should  be  drilled  on 
the  meaning  and  import  of  the  symbols,  until 
their  significations  and  uses  are  as  familiar  as 
its  necessity,  the  sounds  and  combinations  of  the  letters  of  the 
alphabet. 


Elements         §  291.    Beginning  with  the  elements   of  the 

explained.      ,  ,  . 

language,  let  any  number  or  quantity  be  desig 
nated  by  the  letter  a,  and  let  it  be  required  to 


CHAP.   IV.]  ALGEBRA.  273 

add  this  letter  to  itself,  and  find  the  result  or  sum. 
The  addition  will  be  expressed  by 

a  +  a  =  the  sum. 

But  how  is  the  sum  to  be  expressed  ?     By  simply  signification. 
regarding  a  as  one  a,  or  la,  and  then  observing 
that  one  a  and  one  a  make  two  as  or  2  a :  hence, 

a  +  CL  =2a; 

and  thus  we  place  a  figure  before  a  letter  to  in 
dicate  how  many  times  it  is  taken.  Such  figure 
is  called  a  Coefficient.  Coefficient. 

§  292.  The  product  of  several  numbers  is  in-     Product: 
dicated  by  the  sign  of  multiplication,  or  by  sim 
ply  writing  the  letters  which  represent  the  num 
bers  by  the  side  of  each  other.     Thus, 

aXbXGXdxf,  Or  abcdf,  how  indica 

ted. 

indicates  the  continued  product  of  «,  b,  c,  d,  and 
/,  and  each  letter  is  called  a  factor  of  the  prod 
uct  :  hence,  a  factor  of  a  product  is  one  of  the      Factor, 
multipliers  which  produce  it.     Any  figure,  as  5, 
written  before  a  product,  as 

5  abcdf, 

is  the  coefficient  of  the  product,  and  shows  that  coefficient  of 
the  product  is  taken  5  times.  a  product. 

18 


274  MATHEMATICAL     SCIENCE.  [  BOOK  II. 


ten. 


Equal  fac-         §  293.  If  the  numbers  represented  by  a,  b,  c, 
d,  and  /  were  equal  to  each  other,  they  would 
what  the     each  be  represented  by  a  single  letter  a,  and  the 
becomes,    product  would  then  become 

axaxaxaxa  =  a5; 
How       that  is,  we  indicate  the  product  of  several  equal 

expressed. 

factors  by  simply  writing   the   letter  once    arid 
placing  a  figure  above  and  a  little  at  the  right 
of  it,  to  indicate  how  many  times  it  is  taken  as 
Exponent:    a  factor.      The   figure   so  written  is    called   an 
where  writ-  exponent.     Hence,  an  exponent  is  a  simple  form 
of  expression,  to  point  out  how  many  equal  fac 
tors  are  employed. 


Division:         §  294.    The  division  of  one  quantity  by   an- 
how       other  is  indicated  by  simply  writing  the  divisor 
below  the  dividend,  after  the  manner  of  a  frac 
tion  ;  by  placing  it  on  the  right  of  the  dividend 
with  a  horizontal  line  and  two  dots  between  them ; 
or  by  placing  it  on  the  right  with  a  vertical  line 
between  them  :  thus  either  form  of  expression  : 
b 

Three  forms.  ~>     °  +  &>    °r    °  \  a 

indicates  the  division  of  b  by  a. 
Roots:          §  295.  The  extraction  of  a  root  is  indicated 

howteddica"  by the  sisn  V-  This  sisn> when  used  by itself 

indicates  the  lowest  root,  viz.  the  square  root. 


CHAP.   IV.]  ALGEBRA.  275 

If  any  other  root  is  to  be  extracted,  as  the  third, 
fourth,  fifth,  &c.,  the  figure  marking  the  degree      index; 
of  the  root  is  written  above  and  at  the  left  of  where  writ- 

xl         .  ten. 

the  sign ;  as, 

tf~  cube  root,  tf~  fourth  root,  &c. 

The  figure  so  written,  is  called  the  Index  of  the 
root. 

We  have  thus  given  the  very  simple  and  gen-    Language 
eral  language  by  which  we  indicate  every  one   operations! 
of  the  five  operations  that  may  be  performed  on 
an  algebraic  quantity,  and  every  process  in  Al 
gebra  involves  one  or  other  of  these  operations. 


MINUS     S  IGN. 

§  296.  The  algebraic  symbols  are  divided  into    Algebraic 

language : 

two  classes  entirely  distinct   from   each   other, 

viz.   the  letters  that  are  used  to  designate  the  how  divided. 

quantities  which  are  the  subjects  of  the  science, 

and  the  signs  which  are  employed  to  indicate 

certain    operations    to   be   performed   on   those 

quantities.     We  have  seen  that  all  the  algebraic    Algebraic 

processes: 

processes  are  comprised  under  addition,  subtrac- 

their  num- 

tion,  multiplication,  division,  and  the  extraction       ber. 
of  roots  ;  and  it  is  plain,   that  the   nature  of  a      DO  not 
quantity  is  not  at  all  changed  by  prefixing  to  it  ^uT^rine 
the  sign  which  indicates  either  of  these  opera-   ^uantities- 


276  MATHEMATICAL     SCIENCE.  [fiOOK  II. 

tions.  The  quantity  denoted  by  the  letter  a,  for 
example,  is  the  same,  in  every  respect,  whatever 
sign  may  be  prefixed  to  it ;  that  is,  whether  it 
be  added  to  another  quantity,  subtracted  from 
it,  whether  multiplied  or  divided  by  any  number, 
or  whether  we  extract  the  square  or  cube  or  any 
Algebraic  other  root  of  it.  The  algebraic  signs,  therefore, 

signs : 

how  regard-  must  be  regarded   merely  as   indicating   opera 
tions  to  be  performed  on  quantity,   and  not  as 
affecting  the  nature  of  the  quantities  to  which 
they  may  be  prefixed.      We   say,   indeed,   that 
piu3  and     quantities  are  plus  and  minus,  but  this  is  an  ab- 
Mmua.      breviated  language  to  express  that  they  are  to 
be  added  or  subtracted. 

Principles  of  §  297.  In  Algebra,  as  in  Arithmetic  and  Ge 
ometry,  all  the  principles  of  the  science  are  de- 

Fromwhat   duced  from  the  definitions  and  axioms ;  and  the 

deduced.  ru|eg  £or  performing  the  operations  are  but  di 
rections  framed  in  conformity  to  such  principles. 

Example.  Having,  for  example,  fixed  by  definition,  the  power 
of  the  minus  sign,  viz.  that  any  quantity  before 
which  it  is  written,  shall  be  regarded  as  to  be 

what  we  subtracted  from  another  quantity,  we  wish  to 
discover  the  process  of  performing  that  subtrac 
tion,  so  as  to  deduce  therefrom  a  general  prin 
ciple,  from  which  we  can  frame  a  rule  applicable 
to  all  similar  cases. 


a— c 


CHAP.  IV.]  ALGEBRA.  277 


SUBTRACTION. 

§  298.    Let  it   be  required,    for   example,   to  subtraction, 
subtract  from  b  the  difference   be-          7 

Process. 

tween  a  and  c.  i\ow,  having  writ 
ten  the  letters,  with  their  proper 
signs,  the  language  of  Algebra  expresses  that  it 
is  the  difference  only  between  a  and  c,  which  is 
to  be  taken  from  b ;  and  if  this  difference  were  Difference, 
known,  we  could  make  the  subtraction  at  once. 
But  the  nature  and  generality  of  the  algebraic 
symbols,  enable  us  to  indicate  operations,  merely,  operations 

indicated. 

and  we  cannot  in  general  make  reductions  until 
we  come  to  the  final  result.  In  what  general 
way,  therefore,  can  we  indicate  the  true  differ 
ence  ? 


If  we  indicate  the  subtraction  of 
a  from  b,  we  have  b  —  a ;  but  then 


b-a 
b-a 


Final 
formula. 


we  have  taken  away  too  much  from 
b  by  the  number  of  units  in  c,  for  it  was  not  a, 
but  the  difference  between  a  and  c  that  was  to 
be  subtracted  from  b.  Having  taken  away  too 
much,  the  remainder  is  too  small  by  c :  hence, 
if  c  be  added,  the  true  remainder  will  be  express 
ed  by  b  —  a  +  c. 

Now,  by  analyzing  this  result,  we  see  that  the   Analysis  of 

the  result. 

sign  of  every  term  of  the  subtrahend  has  been 
changed ;    and  what  has   been  shown  with  re- 


278  MATHEMATICAL     SCIENCE.  [BOOK  II. 

Generaiiza-  spect  to  these  quantities  is  equally  true  of  all 
others  standing  in  the  same  relation :  hence,  we 
have  the  following  general  rule  for  the  subtrac 
tion  of  algebraic  quantities  : 

Change  the  sign  of  every  term  of  the  subtra- 
Rule-       hend,  or  conceive  it  to  be  changed,  and  then  unite 
the  quantities  as  in  addition. 


MULTIPLICATION . 

Multiplies-        §  299.  Let  us  now  consider  the  case  of  mul- 

tion. 

tiplication,   and  let   it  be   required  to   multiply 

a  —  b  by  c.     The  algebraic  language  expresses 

signification  that  the  difference  between  a  and  b 


of  the 


a  —  b 


ac — be 


language,  is  to  be  taken  as  many  times  as 
there  are  units  in  c.  If  we  knew 
this  difference,  we  could  at  once 
perform  the  multiplication.  But  by  what  gen- 
Process:  eral  process  is  it  to  be  performed  without  finding 
that  difference  ?  If  we  take  a,  c  times,  the  prod 
uct  will  be  ac ;  but  as  it  was  only  the  difference 
between  a  and  b,  that  was  to  be  multiplied  by  c, 
its  nature,  this  product  ac  will  be  too  great  by  b  taken  c 
times ;  that  is,  the  true  product  will  be  expressed 
by  ac  —  be :  hence,  we  see,  that, 

Principle  for  If  a  quantity  having  a  plus  sign  be  multi 
plied  by  another  quantity  having  also  a  plus 
sign,  the  sign  of  the  product  will  be  plus  ;  and 


CHAP.  IV.]  A.LGEBRA.  279 


if  a  quantity  having  a  minus  sign  be  multi 
plied  by  a  quantity  having  a  plus  sign,  the  sign 
of  the  product  will  be  minus. 

§  300.    Let   us  now  take   the   most   general  General  case: 
case,  viz.  that  in  which  it  is  required  to  multi 
ply  a  —  b  by  c  —  d. 

Let  us  again  observe  that  the  algebraic  lan 
guage  denotes  that  a  —  b  is 
to  be  taken  as  many  times 
as  there  are  units  in  c—d\ 


ac — be 
and  if  these  two  differences 


a-b 

c — d  Its  form- 


ac  —  bc  —  ad-\-  bd 
were  known,  their  product     | 

would  at  once  form  the  product  required. 

First :  let  us  take  a  —  b  as  many  times  as  there   First  step. 
are  units   in  c  ;  this  product,  from  what  has  al 
ready  been  shown,   is  equal  to  ac  —  be.      But 
since  the  multiplier  is  not  c,  but  c  —  d,  it  follows 
that  this  product  is  too  large  by  a  —  b  taken  d 
times  ;  that  is,  by  ad  —  bd:  hence,  the  first  prod-  second  step: 
uct  diminished  by  this  last,  will  give  the  true 
product.     But,  by  the   rule  for  subtraction,  this 
difference  is  found  by  changing  the  signs  of  the  Howtaken- 
subtrahend,  and  then  uniting  all  the  terms  as  in 
addition:  hence,  the   true  product  is  expressed 
by  ac  —  be  —  ad  +  bd. 

By   analyzing  this    result,    and  employing   an    Analysis  of 
abbreviated  language,  we  have  the  following  gen- 


280 


MATHEMATICAL     SCIENCE.  [BOOK   II. 


eral  principle  to  which  the  signs  conform  in  mul 
tiplication,  viz. : 

Plus  multiplied  by  plus  gives  plus  in  the  prod 
uct  ;  plus  multiplied  by  minus  gives  minus  ;  mi 
nus  multiplied  by  plus  gives  minus  ;  and  minus 
multiplied  by  minus  gives  plus  in  the  product. 


General 
Principle. 


Remark. 


Particular 
case. 


Minus  sign : 


Its  interpre 
tation. 


Form  of  the 

product : 
must  be  true 
for  quantities 
of  any  value. 


§  301.  The  remark  is  often  made  by  pupils 
that  the  above  reasoning  appears  very  satisfac 
tory  so  long  as  the  quantities  are  presented  un 
der  the  above  form  ;  but  why  will  —b  multiplied 
by  —  d  give  plus  bd  ?  How  can  the  product  of 
two  negative  quantities  standing  alone  be  plus  ? 

In  the  first  place,  the  minus  sign  being  pre 
fixed  to  b  and  d,  shows  that  in  an  algebraic  sense 
they  do  not  stand  by  themselves,  but  are  con 
nected  with  other  quantities ;  and  if  they  are 
not  so  connected,  the  minus  sign  makes  no  dif 
ference ;  for,  it  in  no  case  affects  the  quantity, 
but  merely  points  out  a  connection  with  other 
quantities.  Besides,  the  product  determined 
above,  being  independent  of  any  particular  value 
attributed  to  the  letters  a,  b,  c,  and  d,  must  bo 
of  such  a  form  as  to  be  true  for  all  values ;  and 
hence  for  the  case  in  which  a  and  c  are  both 
equal  to  zero.  Making  this  supposition,  the 
product  reduces  to  the  form  of  +  bd.  The  rules 
for  the  signs  in  division  are  readily  deduced  from 


CHAP.   IV.]  ALGEBRA.  281 


the  definition  of  division,  and  the  principles  al-     signs  m 

j      i'ii  division. 

ready  laid  down. 


ZERO     AND     INFINITY. 

§  302.  The  terms  zero  and  infinity  have  given  zero  and 
rise  to  much  discussion,  and  been  regarded  as 
presenting  difficulties  not  easily  removed.  It  may 
not  be  easy  to  frame  a  form  of  language  that  shall 
convey  to  a  mind,  but  little  versed  in  mathe 
matical  science,  the  precise  ideas  which  these  Ideasnot 

abstruse. 

terms  are  designed  to  express  ;  but  we  are  un 
willing  to  suppose  that  the  ideas  themselves  are 
beyond  the  grasp  of  an  ordinary  intellect.  The 
terms  are  used  to  designate  the  two  limits  of 
Space  and  Number. 


§  303.  Assuming  any  two  points  in  space,  and 
joining  them  by  a  straight  line,  the  distance  be 
tween  the  points  will  be  truly  indicated  by  the 
length  of  this  line,  and  this  length  may  be  ex 
pressed  numerically  by  the  number  of  times 
which  the  line  contains  a  known  unit.  If  now, 
the  points  are  made  to  approach  each  other,  the  lustration, 

,1         c       i         T  MI      i-      •     •   i  showing  the 

length  of    the  line  will  diminish   as  the  points   meaning  of 
come  nearer  and  nearer  together,  until  at  length,       %££* 
when  the  two  points  become  one,  the  length  of 
the  line  will  disappear,  having  attained  its  limit, 


282  MATHEMATICAL     SCIENCE.  [BOOK   II. 

which  is   called  zero.     If,  on   the  contrary,  the 

points  recede  from  each  other,  the  length  of  the 

illustration,  Jine  joining  them  will  continually  increase  ;  but 

showing  the 

meaning  of  so  long  as  the  length  of  the  line  can  be  expressed 

the  term       .  •        /•  •      • 

infinity.  m  terms  oi  a  known  unit  01  measure,  it  is  not 
infinite.  But,  if  we  suppose  the  points  removed, 
so  that  any  known  unit  of  measure  would  occupy 
no  appreciable  portion  of  the  line,  then  the  length 
of  the  line  is  said  to  be  Infinite. 

§  304.  Assuming  one  as  the  unit  of  number, 

and  admitting  the  self-evident  truth  that  it  may 

be   increased   or  diminished,  we    shall  have   no 

zero  andbL  difficulty   in   understanding   the   import    of   the 


terms  zero  and  infinity,  as  applied  to  number. 
For,  if  we  suppose  the  unit  one  to  be  continually 
diminished,  by  division  or  otherwise,  the  frac- 

niustration.  tional  units  thus  arising  will  be  less  and  less, 
and  in  proportion  as  we  continue  the  divisions, 
they  will  continue  to  diminish.  Now,  the  limit 
or  boundary  to  which  these  very  small  fractions 
Zero:  approach,  is  called  Zero,  or  nothing.  So  long 
as  the  fractional  number  forms  an  appreciable 
part  of  one,  it  is  not  zero,  but  a  finite  fraction  ; 
and  the  term  zero  is  only  applicable  to  that 
which  forms  no  appreciable  part  of  the  standard. 

Diustration.  If,  on  the  other  hand,  we  suppose  a  number 
to  be  continually  increased,  the  relation  of  this 


CHAP.   IV.]  ALGEBRA.  283 

number  to  the  unit  will  be  constantly  changing. 
So  long  as  the  number  can  be  expressed  in 
terms  of  the  unit  one,  it  is  finite,  and  not  infi-  infinity; 
nite;  but  when  the  unit  one  forms  no  appre 
ciable  part  of  the  number,  the  term  infinite  is 
used  to  express  that  state  of  value,  or  rather, 
that  limit  of  value. 

§  305.  The  terms  zero  and  infinity  are  there-   The  terms, 
fore  employed  to  designate  the  limits  to  which    employed, 
decreasing   and   increasing    quantities    may   be 
made  to    approach   nearer  than   any  assignable 
quantity ;  but  these  limits  cannot  be  compared,    Are  Iimit3. 
in  respect  to  magnitude,  with  any  known  stand 
ard,  so  as  to  give  a  finite  ratio. 

§  306.  It  may,  perhaps,  appear  somewhat  par-  Whylimite? 
adoxical,  that  zero  and  infinity  should  be  defined 
as  "  the  limits  of  number  and  space"  when  they 
are  in  themselves  not  measurable.     But  a  limit 
is  that  "  which  sets  bounds  to,  or  circumscribes ;"  Definition  of 
and  as   all  finite  space  and  finite  number  (and 
such  only  are  implied  by  the  terms  Space  and  or  space  and 
Number),   are  contained  between  zero  and  in 
finity,  we  employ  these  terms  to  designate   the 
limits  of  Number  and  Space. 


284  MATHEMATICAL     SCIENCE.  [fiOOK  II. 


OF     THE     EQUATION. 

Deductive        §  307.  We  have  seen  that  all  deductive  rea- 

reasoning. 

sonmg  involves  certain  processes  of  comparison, 

and  that  the  syllogism  is  the  formula  to  which 

those  processes  may  be  reduced.*     It  has  also 

comparison  been  stated  that  if  two  quantities  be  compared 

of  quantities. 

together,  there  will  necessarily  result  the  condi- 
tion  of  equality  or  inequality.  The  equation  is 
an  analytical  formula  for  expressing  equality. 


subject  of        §  30g     The   subject  of  equations  is  divided 

equations  : 

how  divided,  into  two  parts.  The  first,  consists  in  finding 
First  part:  the  equation  ;  that  is,  in  the  process  of  express 

ing  the  relations  existing  between  the  quantities 

considered,  by  means  of  the  algebraic   symbols 

statement,    and  formula.      This  is  called  the  Statement  of 

second  part:  the  proposition.     The    second  is  purely  deduc 

tive,  and  consists,  in  Algebra,  in  what  is  called 
Solution,  the  solution  of  the  equation,  or  finding  the  value 

of  the   unknown    quantity  ;    and   in   the    other 

branches  of  analysis,  it  consists  in  the  discus- 
Discussion  of  sion  of  the  equation  ;  that  is,  in  the  drawing  out 

from  the  equation  every  thing  which  it  is  ca 

pable  of  expressing. 

*  Section  98. 


CHAP.    IV.]  ALGEBRA.  285 


§  309.  Making  the  statement,  or  finding  the   statement: 
equation,  is  merely  analyzing  the  problem,  and    what  it  is. 
expressing   its   elements   and   their  relations   in 
the   language  of  analysis.      It  is,  in  truth,  col 
lating  the  facts,  noting  their  bearing  and  con 
nection,  and  inferring  some  general  law  or  prin 
ciple  which  leads  to  the  formation  of  an  equation. 

The  condition  of  equality  between  two  quan-    Equality  of 
tities  is  expressed  by  the  sign  of  equality,  which    w°t^! 
is  placed  between  them.     The  quantity  on  the     HOW  ex- 
left  of  the  sign  of  equality  is  called  the  first  mem-  lgt  member 
ber,  and  that  on  the  right,  the  second  member  2d  member, 
of  the  equation.     The  first  member  corresponds 
to  the  subject  of  a  proposition ;  the  sign  of  equal-     subject, 
ity  is  copula  and  part  of  the  predicate,  signify-    Predicate, 
ing,  is  EQUAL  TO.     Hence,  an  equation  is  merely 
a  proposition   expressed  algebraically,  in  which  Proposition. 
equality  is  predicated  of  one  quantity  as  com 
pared  with  another.     It  is   the  great  formula  of 
analysis. 

§  310.  We  have  seen  that  every  quantity  is    Abstract. 
either  abstract  or  concrete  :*    hence,  an  equa-    concrete, 
tion,  which  is  a  general  formula  for  expressing 
equality,  must  be  either  abstract  or  concrete. 

An   abstract   equation   expresses   merely    the 

*  Section  75. 


286 


MATHEMATICAL     SCIENCE.  [BOOK  II. 


relation  of  equality  between  two  abstract  quan 
tities  :  thus, 


Abstract 
equation. 


is  an  abstract  equation,  if  no  unit  of  value  be 
assigned  to  either  member ;  for,  until  that  be 
done  the  abstract  unit  one  is  understood,  and  the 
formula  merely  expresses  that  the  sum  of  a  and  b 
is  equal  to  x,  and  is  true,  equally,  of  all  quantities, 
concrete  But  if  we  assign  a  concrete  unit  of  value,  that 
is,  say  that  a  and  b  shall  each  denote  so  many 
pounds  weight,  or  so  many  feet  or  yards  of 
length,  x  will  be  of  the  same  denomination,  and 
the  equation  will  become  concrete  or  denominate. 

rive  opera-       §  311.  We  have  seen  that  there  are  five  oper- 

tions  maybe 

performed,  ations  which  may  be  performed  on  an  algebraic 
quantity.*  We  assume,  as  an  axiom,  that  if 
the  same  operation,  under  either  of  these  pro 
cesses,  be  performed  on  both  members  of  an 
equation,  the  equality  of  the  members  will  not  be 
changed.  Hence,  we  have  the  five  following 


Axioms. 


First. 


AXIOMS. 

1.  If  equal  quantities  be  added  to  both  mem 
bers  of  an  equation,  the  equality  of  the  members 
will  not  be  destroyed. 

*  Section  288. 


CHAP.  IV.] 


ALGEBRA. 


287 


2.  If  equal  quantities  be  subtracted  from  both     second, 
members  of  an  equation,  the  equality  will  not  be 
destroyed. 

3.  If  both  members  of  an  equation  be  multi-      Third, 
plied  by  the  same  number,  the  equality  will  not 

be  destroyed. 

4.  If  both  members  of  an  equation  be  divided 
by  the  same  number,   the  equality  will  not  be 
destroyed. 

5.  If  the   same  root  of  both  members  of  an 
equation  be  extracted,  the  equality  of  the  mem 
bers  will  not  be  destroyed. 

Every  operation  performed  on  an  equation 
will  fall  under  one  or  other  of  these  axioms,  and 
they  afford  the  means  of  solving  all  equations 
which  admit  of  solution. 


Fourth. 


Fifth. 


Use  of 

axioms. 


§  312.  The  term  Equality,  in  Geometry,  ex-     Equality: 

Its  meaning 

presses   that   relation  between   two   magnitudes  in  Geometry. 

which  will  cause  them  to  coincide,  throughout 

their  whole  extent,  when  applied  to  each  other. 

The  same  term,  in  Algebra,  merely  implies  that  its  meaning 

in  Algebra. 

the  quantity,  of  which  equality  is  predicated, 
and  that  to  which  it  is  affirmed  to  be  equal, 
contain  the  same  unit  of  measure  an  equal  num 
ber  of  times  :  hence,  the  algebraic  signification 
of  the  term  equality  corresponds  to  the  signi-  corresponds 

to  equiva- 

fication  of  the  geometrical  term  equivalency.  lency. 


288  MATHEMATICAL     SCIENCE.  [BOOK   II. 

§  313.  We  have  thus  pointed  out  some  of  the 

marked  characteristics  of  analysis.     In  Algebra, 

classes  of    the    elementary   branch,    the    quantities,    about 

quantities  in  .  .     . 

Algebra,  which  the  science  is  conversant,  are  divided, 
as  has  been  already  remarked,  into  known  and 
unknown,  and  the  connections  between  them, 
expressed  by  the  equation,  afford  the  means  of 
tracing  out  further  relations,  and  of  finding  the 
values  of  the  unknown  quantities  in  terms  of  the 
known. 

In  the  other  branches  of  analysis,  the  quanti- 
HOW  divided  ties    considered    are    divided    into    two   general 

in  the  other 

branches  of  classes,  Constant  and  Variable  ;  the  former  pre- 

Analysis.  .  /•        i          i 

serving  fixed  values  throughout  the  same  pro 
cess  of  investigation,  while  the  latter  undergo 
changes  of  value  according  to  fixed  laws;  and 
from  such  changes  we  deduce,  by  means  of  the 
equation,  common  principles,  and  general  prop 
erties  applicable  to  all  quantities. 

correspond-       §  314.  The  correspondence  between  the  pro- 

ence  in 

methods  of  cesses  of  reasoning,  as  exhibited  in  the  subjectvof 
accounted    general  logic,  and  those  which  are  employed  in 
for'        mathematical  science,  is  readily  accounted  for, 
when  we  reflect,  that  the  reasoning  process  is 
essentially  the  same  in  all  cases ;  and  that  any 
change  in  the  language  employed,  or  in  the  sub 
ject  to  which  the  reasoning  is  applied,  does  not 


CHAP.   IV.]  ALGEBRA.  289 

at  all  change  the  nature  of  the  process,  or  mate 
rially  vary  its  form. 

§  315.  We  shall  not  pursue  the  subject  of 
analysis  any  further;  for,  it  would  be  foreign 
to  the  purposes  of  the  present  work  to  attempt  objects  of 

,     ,,  ,     the  present 

more  than  to  point  out  the  general  features  and      work: 
characteristics  of  the  different  branches  of  math 
ematical  science,  to  present  the  subjects  about 
which  the  science  is  conversant,  to  explain  the 
peculiarities  of  the  language,  the  nature  of  the 
reasoning  processes  employed,  and  of  the  con 
necting  links  of  that  golden  chain  which  binds    extended, 
together  all  the  parts,   forming  an   harmonious 
whole. 


SUGGESTIONS  FOR  THOSE  WHO  TEACH  ALGEBRA. 

1.  Be  careful  to  explain  that  the  letters  em-   Letters  are 
ployed,  are  the  mere  symbols  of  quantity.     That     8ym™^ 
of,  and  in  themselves,  they  have  no  meaning  or 
signification  whatever,  but  are  used  merely  as 

the  signs  or  representatives  of  such  quantities 
as  they  may  be  employed  to  denote. 

2.  Be  careful  to  explain  that  the  signs  which   signs  indi- 

,,          .  cate  opera- 

are  used  are  employed  merely  for  the  purpose       tions. 

of  indicating  the  five  operations  which  may  be 

performed   on  quantity ;  and  that  they  indicate 

19 


SCO  MATHEMATICAL     SCIENCE.  [BOOK  II. 

operations  merely,  without  at  all  affecting  the 
nature  of  the  quantities  before  which  they  are 
placed. 

Letters  and  3.  Explain  that  the  letters  and  signs  are  the 
eieme  's  of  elements  °f  tne  algebraic  language,  and  that  the 
language,  language  itself  arises  from  the  combination  of 

these  elements. 

Algebraic        4.  Explain  that    the  finding  of  an  algebraic 
formula  is  but  the  translation  of  certain  ideas, 
first  expressed   in  our  common   language,   into 
the  language  of  Algebra ;  and  that  the  interpre 
ts  interpret-  tation  of  an   algebraic  formula  is  merely  trans 

ation. 

lating  its  various  significations  into  common 
language. 

Language.  5.  Let  the  language  of  Algebra  be  carefully 
studied,  so  that  its  construction  and  significa 
tions  may  be  clearly  apprehended. 

coefficient,       6.    Let  the   difference  between  a  coefficient 
Exponent    and  an  exponent   be   carefully  noted,    and   the 
office  of  each  often  explained ;  and  illustrate  ire 
quently  the  signification  of  the  language  by  at 
tributing  numerical  values  to  letters  in  various 
algebraic  expressions, 
similar         7.  Point  out  often  the  characteristics  of  sim- 

quantities. 

ilar  and  dissimilar  quantities,  and  explain  which 
may  be  incorporated  and  which  cannot. 
Minus  sign.       Q.  Explain  the  power  of  the  minus  sign,  as 
shown  in  the  four  ground  rules,  but  very  par- 


CHAP.  IV.]  ALGEBRA.  291 

ticularly  as  it  is  illustrated  in  subtraction  and 
multiplication. 

9.  Point  out  and  illustrate  the  correspondence 
between   the  four   ground  rules   of  Arithmetic    Arithmetic 

and  Algebra 

and  Algebra;  and  impress  the  fact,  that  their    compared, 
differences,  wherever  they  appear,  arise  merely 
from  differences  in  notation  and  language  :  the 
principles  which   govern   the    operations   being 
the  same  in  both. 

10.  Explain  with  great  minuteness    and  par-    Equation, 
ticularity  all  the  characteristic  properties  of  the    its  proper 

ties. 

equation ;  the  manner  of  forming  it ;  the  differ 
ent  kinds  of  quantity  which  enter  into  its  com 
position  ;  its  examination  or  discussion  ;  and 
the  different  methods  of  elimination. 

11.  In  the  equation  of  the  second  degree,  be  Equation  oi 
careful  to  dwell  on  the  four  forms  which  em 


brace  all  the  cases,  and  illustrate  by  many  ex 
amples  that  every  equation  of  the  second  de 
gree  may  be  reduced  to  one  or  other  of  them,     its  forms. 
Explain   very  particularly  the   meaning  of  the 
term  root ;  and  then  show,  why  every  equation     Its  roots- 
of  the  first  degree  has  one,  and  every  equation 
of  the  second  degree  two.     Dwell  on  the  prop 
erties  of  these  roots  in  the  equation  of  the  sec 
ond  degree.     Show  why  their   sum,  in  all  the   Their  sum. 
forms,  is  equal  to  the  coefficient  of  the  second 
term,  taken  with  a  contrary  sign ;  and  why  their 


292  MATHEMATICAL     SCIENCE.  [BOOK  II. 


Their  prod-  product  is    equal  to  the  absolute  term  with  a 
uct'       contrary  sign.     Explain  when  and  why  the  roots 

are  imaginary. 
General          \2.    In  fine,   remember  that  every  operation 

Principles:  .  .......  , 

and  rule  is  based  on  a  principle  of  science,  and 
that  an  intelligible  reason  may  be  given  for  it. 
Find  that  reason,  and  impress  it  on  the  mind 
should  be  of  your  pupil  in  plain  and  simple  language,  and 
by  familiar  and  appropriate  illustrations.  You 
will  thus  impress  right  habits  of  investigation 
and  study,  and  he  will  grow  in  knowledge.  The 
broad  field  of  analytical  investigation  will  be 
opened  to  his  intellectual  vision,  and  he  will 
have  made  the  first  steps  in  that  sublime  science 
They  lead  to  which  discovers  the  laws  of  nature  in  their  most 

general  lawa. 

secret  hiding-places,  and  follows  them,  as  they 
reach  out,  in  omnipotent  power,  to  control  the 
motions  of  matter  through  the  entire  regions  of 
occupied  space. 


BOOK    III. 

UTILITY    OF   MATHEMATICS, 


CHAPTER   I. 

THE   UTILITY    OF    MATHEMATICS    CONSIDERED    AS    A    MEANS    OF    INTELLECTUAL 
TRAINING    AND    CULTURE. 

§  316.    THE  first  efforts  in  mathematical  sci-  Firet  efforts- 
ence  are  made  by  the  child  in  the   process  of 
counting.      He  counts  his  fingers,   and  repeats 
the  words  one,  two,  three,  four,  five,  six,  seven,   * 
eight,  nine,  ten,  until  he  associates  with  these       jects. 
words  the  ideas  of  one  or  more,  and  thus  ac 
quires  his  first  notions  of  number.     Hence,  the 
idea  of  number  is  first  presented  to  the  mind  by 
means  of  sensible  objects ;  but  when  once  clear 
ly  apprehended,  the  perception  of  the  sensible 
objects  fades  away,   and  the  mind  retains   only 
the  abstract  idea.     Thus,  the  child,  after  count-    General- 

tion. 

ing  for  a  time  with  the  aid  of  his  fingers  or  his 
marbles,  dispenses  with  these  cumbrous  helps,  and 


294  UTILITY     OF     MATHEMATICS.  [BOOK  III. 

Abstraction,  employs  only  the  abstract  ideas,  which  his  mind 
embraces  with  clearness  and  uses  with  facility. 

Analytical        §  317.    In  the  first  stages  of  the  analytical 

method:  ,  ,  .   .  •  i          i 

methods,   where   the   quantities   considered   are 

uses  sensible  represented  by  the  letters  of  the   alphabet,  sen- 

°  first!3      sible  objects  again  lend  their  aid  to  enable  the 

mind  to  gain  exact   and  distinct   ideas  of    the 

things  considered ;  but  no  sooner  are  these  ideas 

obtained  than  the  mind  loses  sight  of  the  things 

themselves,   and   operates   entirely   through   the 

instrumentality  of  symbols. 

Geometry.         §  318.  So,  also,  in  Geometry.     The  right  line 

may  first  be  presented  to  the  mind,  as  a  black 

First  impres-  mark  on  paper,  or  a  chalk  mark  on  a  black- 

sions  by  sen-  .  ,          .,     .   . 

sibie  objects,  board,  to  impress  the  geometrical  definition,  that 
"  A  straight  line  does  not  change  its  direction 
between  any  two  of  its  points."  When  this 
definition  is  clearly  apprehended,  the  mind  needs 
no  further  aid  from  the  eye,  for  the  image  is 
forever  imprinted. 

A  plane.         §  319.  The  idea  of  a  plane  surface  may  be 

Definition:    impressed  by  exhibiting  the  surface  of  a  polished 

mirror;    and   thus   the  mind   may  be   aided  in 

HOW  mustra-  apprehending  the  definition,  that  "  a  plane  sur- 

ted. 

face  is  one  in  which,  if  any  two  points  be  taken, 


CHAP.  I.]  QUANTITY SPACE.  295 

the  straight  line  which  joins  them  will  lie  wholly 
in  the  surface."  But  when  the  definition  is 
understood,  the  mind  requires  no  sensible  object  Itetrue 

conception. 

to  aid  its  conception.  The  ideal  alone  fills  the 
mind,  and  the  image  lives  there  without  any 
connection  with  sensible  objects. 

§  320.  Space  is  indefinite  extension,  in  which      space, 
all  bodies  are  situated.     A  solid  or  body  is  any      Solid: 
portion  of  space  embracing  the  three  dimensions 
of  length,  breadth,  and  thickness.     To  give  to 

the  mind  the  true  conception  of  a  solid,  the  aid    HOW  con 
ceived, 
of  the  eye  may  at  first  be  necessary;  but  the 

idea  being  once  impressed,  that  a  solid,  in  a 
strictly  mathematical  sense,  means  only  a  por 
tion  of  space,  and  has  no  reference  to  the  mat-  what  it 
ter  with  which  the  space  may  be  filled,  the  mind 
turns  away  from  the  material  object,  and  dwells 
alone  on  the  ideal. 

§  321.  Although  quantity,  in  its  general  sense,  Quantity: 
is  the  subject  of  mathematical  inquiry,  yet  the 

language  of  mathematics  is  so  constructed,  that  Language: 

the  investigations  are  pursued  without  the  slight-  HOW  con- 

structed. 

est  relerence  to  quantity  as  a  material  substance. 
We  have  seen  that  a  system   of   symbols,  by 
which  quantities  may  be  represented,  has  been    symbols: 
adopted,  forming  a  language  for  the  expression 


296  UTILITY     OF     MATHEMATICS.  [BOOK  III. 

of  ideas  entirely  disconnected  from  material  ob 
jects,  and  yet  capable   of  expressing  and  repre- 
Nitureof    senting  such  objects.     This  symbolical  language, 

the  lan 
guage:      at  once  copious  and  exact,  not  only  enables  us 

to  express  our  known  thoughts,  in  every  depart- 
whatitac-  ment  of  mathematical  science,  but  is  a  potent 

complishes.  ~  ,  .  ....  ,          , 

means  of  pushing  our  inquiries  into  unexplored 
regions,  and  conducting  the  mind  with  certainty 
to  new  and  valuable  truths. 

Advantages  §  322.  The  nature  of  that  culture,  which  the 
cact  mind  undergoes  by  being  trained  in  the  use  of 
an  exact  language,  in  which  the  connection  be 
tween  the  sign  and  the  thing  signified  is  unmis 
takable,  has  been  well  set  forth  by  a  living 
author,  greatly  distinguished  for  his  scientific 
attainments.*  Of  the  pure  sciences,  he  says 

Herschei's  "  Their  objects  are  so  definite,  and  our  no- 
3W8<  tions  of  them  so  distinct,  that  we  can  reason 
about  them  with  an  assurance  that  the  words  and 
signs  of  our  reasonings  are  full  and  true  repre 
sentatives  of  the  things  signified ;  and,  conse- 

Exactian-  quently,  that  when  we  use  language  or  signs  in 
'..  argument,  we  neither  by  their  use  introduce 
extraneous  notions,  nor  exclude  any  part  of  the 
case  before  us  from  consideration.  For  exarn- 

*  Sir  John  Herschel,  Discourse  on  the  study  of  Natural 
Philosophy. 


CHAP.  I.]  EXACT     TERMS.  297 

pie  :  the  words  space,  square,  circle,  a  hundred,  Mathematical 

terms  exact. 

&c.,  convey  to  the  mind  notions  so  complete 
in  themselves,  and  so  distinct  from  every  thing 
else,  that  we  are  sure  when  we  use  them  we 
know  and  have  in  view  the  whole  of  our  own 
meaning.  It  is  widely  different  with  words  ex-  Different  in 

.  .  regard  to 

pressing  natural  objects  and  mixed  relations.  other  terms> 
Take,  for  instance,  Iron.  Different  persons  at 
tach  very  different  ideas  to  this  word.  One  who 
has  never  heard  of  magnetism  has  a  widely  dif 
ferent  notion  of  iron  from  one  in  the  contrary 
predicament.  The  vulgar  who  regard  this  metal  HOW  iron  is 

regarded  by 

as  incombustible,  and  the  chemist,  who  sees  it  the  chemist: 
burn  with  the  utmost  fury,  and  who  has  other 
reasons  for  regarding  it  as  one  of  the  most  com 
bustible  bodies  in   nature ;    the  poet,  who  uses   The  poet  • 
it  as  an  emblem  of  rigidity  ;  and  the  smith  and 
engineer,  in  whose  hands  it  is  plastic,  and  mould 
ed  like  wax  into  every  form ;  the  jailer,  who  prizes   The  jailer: 
it   as   an  obstruction,  and  the   electrician,  who  The  eiectri- 
sees  in  it  only  a  channel  of  open  communication 
by  which  that  most  impassable  of  obstacles,  the 
air,  may  be  traversed  by  his  imprisoned  fluid, — 
have  all  different,  and  all  imperfect  notions  of 
the  same  word.     The  meaning  of  such  a  term   Final  nius- 
is  like  the  rainbow — everybody  sees  a  different 
one,  and  all  maintain  it  to  be  the  same." 

"  It  is,  in  fact,  in  this  double  or  incomplete 


L 


298  UTILITY     OF     MATHEMATICS.          [fiOOK  III. 


incomplete   sense  of  words,  that  we  must  look  for  the  origin 

meaning  the       c  ,  ~     ,  .  ... 

source  of  °»  a  very  large  portion  of  the  errors  into  which 
error-  we  fall.  Now,  the  study  of  the  abstract  sciences, 

Mathematics  such    as   Arithmetic,    Geometry,    Algebra,   &c., 

such  errors,  while  they  afford  scope  for  the  exercise  of  rea 
soning  about  objects  that  are,  or,  at  least,  may 
be  conceived  to  be,  external  to  us;  yet,  being 
free  from  these  sources  of  error  and  mistake, 

Requires  a   accustom   us  to  the  strict   use   of   language  as 

strict  use  of  .  .    . 

language,  an  instrument  of  reason,  and  by  familiarizing  us 
in  our  progress  towards  truth,  to  walk  uprightly 
and  straightforward,  on  firm  ground,  give  us 
that  proper  and  dignified  carriage  of  mind  which 
Results,  could  never  be  acquired  by  having  always  to 
pick  our  steps  among  obstructions  and  loose 
fragments,  or  to  steady  them  in  the  reeling  tem 
pests  of  conflicting  meanings." 

TWO  ways  of      §  323.  Mr.  Locke  lays  down  two  ways  of  in- 
acquiring 
knowledge,   creasing  our  knowledge  : 

1st.  "Clear  and  distinct  ideas  with  settled 
names  ;  and, 

2d.  "  The  finding  of  those  which  show  their 
agreement  or  disagreement ;"  that  is,  the  search 
ing  out  of  new  ideas  which  result  from  the  com 
bination  of  those  that  are  known. 

First.  In  regard  to  the  first  of  these  ways,  Mr.  Locke 

says  :  "  The  first  is  to  get  and  settle  in  our  minds 


CHAP.  I.]  INCREASING     KNOWLEDGE.  299 


determined  ideas  of  those   things,  whereof  we     ideas  of 
have  general  or  specific  names  ;  at  least,  of  so   be  distinct< 
many  of  them  as  we  would  consider  and  im 
prove  our  knowledge  in,  or  reason  about."  *  *  * 
"  For,  it  being  evident,  that  our  knowledge  can 
not  exceed  our  ideas,  as  far  as  they  are  either  im 
perfect,  confused,  or  obscure,  we  cannot  expect 
to  have  certain,  perfect,  or  clear  knowledge." 


§  324.    Now,   the  ideas   which  make  up  our   why  it  is 

.  C  ir-i  so  in  mathe- 

knowledge  oi  mathematical  science,  mini  ex-  matic3. 
actly  these  requirements.  They  are  all  im 
pressed  on  the  mind  by  a  fixed,  definite,  and 
certain  language,  and  the  mind  embraces  them 
as  so  many  images  or  pictures,  clear  and  dis 
tinct  in  their  outlines,  with  names  which  sug 
gest  at  once  their  characteristics  and  properties. 

§  325.    In  the  second  method  of  increasing     second. 
our  knowledge,  pointed  out  by  Mr.  Locke,  math 
ematical  science  offers  the  most  ample  and  the  why  mathe- 

m,  .  11  i  j  matics  offer 

surest  means.     The  reasonings  are  all  based  on    the  suregt 
self-evident  truths,  and  are  conducted  by  means      meaus- 
of  the  most  striking  relations  between  the  known 
and  the  unknown.     The  things  reasoned  about, 
and  the  methods   of   reasoning,    are  so  clearly 
apprehended,  that  the  mind  never  hesitates  01 
doubts.     It  comprehends,  or  it  does  not  compre- 


300  UTILITY     OF     MATHEMATICS.  [BOOK  III. 


hend,  and  the  .  line  which  separates  the  known 
characteris-  from  the  unknown,  is  always  well  defined.  These 

tics  of  the 

reasoning,  characteristics  give  to  this  system  of  reasoning 
itsadvan-  a  superiority  over  every  other,  arising,  not  from 
any  difference  in  the  logic,  but  from  a  difference 
in  the  things  to  which  the  logic  is  applied.  Ob 
servation  may  deceive,  experiment  may  fail,  and 
experience  prove  treacherous,  but  demonstration 


tion  certain. 

never. 
Mathematics       "  If  it  be  true,  then,  that  mathematics  include 

includes  a  ~  r  . 

certain  sys-   a  perfect   system  01  reasoning,  whose  premises 

'em'       are  self-evident,  and  whose  conclusions  are  irre 

sistible,  can  there  be  any  branch  of  science  or 

knowledge  better  adapted  to  the  improvement 

of  the  understanding?     It  is  in   this    capacity, 

An  adjunct   as   a  strong  and  natural  adjunct  and  instrument 

and  instru-         ,,  ...  „ 

ment  of  rea-  °f  reason,  that  this  science  becomes  the  fit  sub-. 
lon;  ject  of  education  with  all  conditions  of  society, 
whatever  may  be  their  ultimate  pursuits.  Most 
sciences,  as,  indeed,  most  branches  of  knowledge, 
address  themselves  to  some  particular  taste,  or 
subsequent  avocation  ;  but  this,  while  it  is  be 
fore  all,  as  a  useful  attainment,  especially  adapts 
itself  to  the  cultivation  and  improvement  of  the 

and  necessa-  thinking  faculty,  and  is  alike  necessary  to  all 
^  who  would  be  governed  by  reason,  or  live  for 
usefulness."* 

*  Mansfield's  Discourse  on  the  Mathematics. 


CHAP.  I.]  REASONS.  301 


§  326.  The  following,  among  other  consider- 
ations,  may  serve  to  point  out  and  illustrate  the     value  of 
value  of  mathematical   studies,  as   a   means   of  mathematics- 
mental  improvement  and  development. 

1.    We  readily  conceive   and  clearly   appre-       First. 

They  give 

hend  the   things  of  which   the   science   treats  ;  clear 


they  being  things  simple  in  themselves  and  read- 

ily  presented  to  the  mind  by  plain  and  familiar 

language.     For  example  :  the  idea  of  number,  of 

one  or  more,  is  among  the  first  ideas  implanted    Example. 

in  the  mind  ;  and  the  child  who  counts  his  fin 

gers  or  his  marbles,  understands  the  art  of  num 

bering  them  as  perfectly  as  he  can  know  any 

thing.    So,  likewise,  when  he  learns  the  definition   They  estab- 

of  a  straight  line,  of  a  triangle,  of  a  square,  of  relations  be- 


a  circle,  or  of  a  parallelogram,  he  conceives  the 

idea  of  each  perfectly,   and  the  name  and  the      thing8- 

image  are  inseparably  connected.     These  ideas, 

so  distinct  and  satisfactory,  are  expressed  in  the 

simplest  and  fewest  terms,  and  may,  if  necessary, 

be  illustrated  by  the  aid  of  sensible  objects. 

2.    The   words   employed   in   the   definitions     second. 

Words  are 

are  always  used  in   the   same  sense  —  each  ex-  always  used 

,,  ,  .  i  ,  i      ,     in  the  same 

pressing  at  all  times  the  same  idea  ;  so  that 
when  a  definition  is  apprehended,  the  concep 
tion  of  the  thing,  whose  name  is  defined,  is  per 
fect  in  the  mind. 

There  is,   therefore,  no   doubt   or   ambiguity 


302  UTILITY     OF     MATHEMATICS.  [BOOK  III. 

Hence,  it  is   either  in  the  language,  or  in  regard  to  what  is 

affirmed  or  denied  of  the  things  spoken  of;  but 

all  is  certainty,  both  in  the  language  employed 

and  in  the  ideas  which  it  expresses. 

Third.          3.    The  science  of  mathematics  employs  no 

no  definition  definition  which   may  not   be   clearly   compre- 

or  axiom  not  i         j    j       i  j 

identand  nended  —  lays  down  no  axioms  not  universally 
true,  and  to  which  the  mind,  by  the  very  laws 
of  its  nature,  readily  assents  ;  and  because,  also, 
in  the  process  of  the  reasoning,  no  principle  or 
truth  is  taken  for  granted,  but  every  link  in 
Theconnec-  the  chain  of  the  argument  is  immediately  con- 

tion  evident.  ,        .  ,  ,    n    .   . 

nected  with  a  definition  or  axiom,  or  with  some 
principle  previously  established. 
Fourth.         4.    The   order   established   in   presenting   the 

The  order  . 

strengthens   subject   to  the   mind,  aids   the   memory   at  the 


ev 

clear. 


same  time  tnat  it  strengthens  and  improves  the 
reasoning   powers.      For   example:    first,    there 
HOW  ideas    are  the  definitions  of  the  names  of  the  things 
ire  presen,-  ^-^  are  ^  subjects  of  the  reasoning;    then 

the    axioms,    or   self-evident    truths,   which,    to 

gether  with  the  definitions,  form  the  basis  of  the 

science.      From  these  the  simplest  propositions 

HOW  the  de-  are  deduced,  and  then  follow  others  of  greater 

iUCio°wSftl"  difficulty;  the  whole  connected  together  by  rig 

orous  logic  —  each  part  receiving  strength    and 

light  from  all  the  others.     Whence,  it  follows, 

that  any  proposition  may  be  traced  to  first  prin- 


CHAP.  I.]  SYNTHESIS ANALYSIS.  303 

ciples ;  its  dependence  upon  and  connection  Propositions 
with  those  principles  made  obvious  ;  and  its  truth  the'iTsources. 
established  by  certain  and  infallible  argument. 

5.    The   demonstrative   argument   of  mathe-      Fifth- 

Argument 

matic^    produces    the   most    certain    knowledge     the  most 

cert  aiii* 

of  which  the  mind  is  susceptible.  It  estab 
lishes  truth  so  clearly,  that  none  can  doubt  or 
deny.  For,  if  the  premises  are  certain — that  is,  Reasons. 
such  that  all  minds  admit  their  truth  without 
hesitation  or  doubt,  and  if  the  method  of  draw 
ing  the  conclusions  be  lawful — that  is,  in  accord 
ance  with  the  infallible  rules  of  logic,  the  infer 
ences  must  also  be  true.  Truths  thus  established 
may  be  relied  on  for  their  verity  ;  and  the  knowl-  such 
edge  thus  gained  may  well  be  denominated 
SCIENCE. 

§  327.  There  are,  as  we  have  seen,  in  mathe 
matics,  two  systems  of  investigation  quite  differ 
ent  from  each  other :    the  Synthetical  and  the    gynthesis, 
Analytical ;  the  synthetical  beginning  with  the    Analysis- 
definitions  and  axioms,  and  terminating  in  the 
highest  truth  reached  by  Geometry. 

"This  science  presents  the  very  method  by  Synthetical 
which   the   human   mind,  in   its   progress  from 
childhood  to  age,  develops  its  faculties.     What 
first  meets  the  observation  of  a  child  ?     Upon  Firgt  notionfl 
what  are  his  earliest  investigations  employed  ? 


304  UTILITY     OF     MATHEMATICS.  [BOOK   III. 


is  first  Next  to  color,  which  exists  only  to  the  sight, 

observed. 

figure,  extension,  dimension,  are  the  first  objects 
which  he  meets,  and  the  first  which  he  examines. 
He  ascertains  and  acknowledges  their  existence  ; 
then  he  perceives  plurality,  and  begins  to  enu- 

Progress  of  merate  ;  finally  he  begins  to  draw  conclusions 
from  the  parts  to  the  whole,  and  makes  a  law 
from  the  individual  to  the  species.  Thus,  he 
has  obtained  figure,  extension,  dimension,  enu 
meration,  and  generalization.  This  is  the  teach 
ing  of  nature  ;  and  hence,  when  this  process 

Process  de-   becomes  embodied  in  a  perfect  system,  as  it  is 

veloped  in     .        ^ 

the  system  of  m   Geometry,   that  system   becomes  the   easiest 
letry'    and  most  natural  means   of  strengthening   the 

mind  in  its  early  progress  through  the  fields  of 

knowledge." 
First  neces-        "  Long  after  the  child  has  thus  begun  to  gen- 

eranze  and  deduce  laws,  he  notices  objects  and 


events,  whose  exterior  relations  afford  no  con 
clusion  upon  the  subject  of  his  contemplation. 
Machinery  is  in  motion — effects  are  produced, 
its  method.   He  is   surprised ;    examines   and  inquires.      He 
reasons  backward  from  effect  to  cause.     This  is 
Analysis,  the  metaphysics  of  mathematics ;  and 
what  the    through  all  its  varieties — in  Arithmetic — in  Alge- 
6cienceis:   bra— and  in  the  Differential  and  Integral  Calcu 
lus,  it  furnishes  a  grand  armory  of  weapons  for 
acute  philosophical  investigation.     But  analysis 


CHAP,  i.]  BACON'S    OPINION.  305 

advances  one  step  further  by  its  peculiar  nota-  what  it  does-. 
tion;  it  exercises,  in  the  highest  degree,  the  fac 
ulty  of  abstraction,  which,  whether  morally  or 
intellectually  considered,  is  always  connected 
with  the  loftiest  efforts  of  the  mind.  Thus  this 
science  comes  in  to  assist  the  faculties  in  their 
progress  to  the  ultimate  stages  of  reasoning; 
and  the  more  these  analytical  processes  are  cul-  what  it 

finally  ac 
tivated,  the  more   the  mind  looks  in  upon  itself,   compiishes. 

estimates  justly  and  directs  rightly  those  vast 
powers  which  are  to  buoy  it  up  in  an  eternity 
of  future  being."* 

§  328.  To  the  quotations,  which  have  already 
been  so  ample,  we  will  add  but  two  more. 

"  In    the   mathematics,   I   can  report  no  defi-      Bacon's 

,  -,        opinion  of 

cience,  except  it  be  that  men  do  not  sum-  mathematics. 
ciently  understand  the  excellent  use  of  the  pure 
mathematics,  in  that  they  do  remedy  and  cure 
many  defects  in  the  wit  and  faculties  intellectual. 
For,  if  the  wit  be  too  dull,  they  sharpen  it ;  if 
too  wandering,  they  fix  it ;  if  too  inherent  in  the 
sense,  they  abstract  it."f  Again  : 

"  Mathematics  serve  to  inure  and  corroborate     HOW  the 

study  of 

the  mind  to  a   constant  diligence  in   study,  to 


*  Mansfield's  Discourses  on  Mathematics, 
f  Lord  Bacon. 

20 


336  UTILITY     OF     MATHEMATICS.  [BOOK  III. 

mathematics  undergo  the  trouble  of  an  attentive  meditation, 
^rnind.       and  cheerfully  contend  with  such  difficulties  as 
lie  in  the  way.     They  wholly  deliver  us  from 
credulous    simplicity,   most    strongly   fortify   us 
against  the  vanity  of  skepticism,  effectually  re- 
its  influences,  strain  us  from  a  rash  presumption,  most  easily 
incline  us  to  due  assent,  perfectly  subjugate  us 
to  the  government  and  weight  of  reason,   and 
inspire  us  with  resolution  to  wrestle  against  the 
injurious  tyranny  of  false  prejudices. 

HOW  they  are  "If  the  fancy  be  unstable  and  fluctuating,  it 
is,  as  it  were,  poised  by  this  ballast,  and  steadied 
by  this  anchor ;  if  the  wit  be  blunt,  it  is  sharp 
ened  by  this  whetstone ;  if  it  be  luxuriant,  it  is 
pruned  by  this  knife ;  if  it  be  headstrong,  it  is 
restrained  by  this  bridle  ;  and  if  it  be  dull,  it  is 
roused  by  this  spur."* 

§  329.  Mathematics,  in  all  its  branches,  is,  in 
fact,  a  science  of  ideas  alone,  unmixed  with  mat- 
Mathematics  ter  or  material  things;   and  hence,  is  properly 
a  pure  sci-    terme(j  a  pure  Science.     It  is,  indeed,  a  fairy 

ence. 

land  of  the  pure  ideal,  through  which  the  mind 
is  conducted  by  conventional  symbols,  as  thought 
is  conveyed  along  wires  constructed  by  the 
hand  of  man. 

*  Dr.  Barrow. 


CHAP.  I."]  CONCLUSION  307 

§  330.  In  conclusion,  therefore,  we  may  claim   what  may 
for  the  study  of  Mathematics,  that  it  impresses   cilhnedfor 
the   mind  with  clear  and  distinct   ideas ;  culti-  m 
vates  habits  of  close  and  accurate   discrimina 
tion  ;    gives,  in  an  eminent  degree,  the  power 
of  abstraction ;  sharpens  and  strengthens  all  the 
faculties,   and  develops,  to  their  highest  range, 
the  reasoning   powers.      The   tendency  of  this  its  tendency, 
study  is  to  raise  the  mind  from  the  servile  habit 
of  imitation  to  the  dignity  of  self-reliance  and 
self-action.     It  arms  it  with  the  inherent  ener 
gies  of  its  own  elastic  nature,  and  urges  it  out  The  reasons. 
on  the  great  ocean   of  thought,  to  make  new 
discoveries,  and  enlarge  the  boundaries  of  men 
tal  effort. 


ICS  UTILITY     OF     MATHEMATICS.  [BOOK   III. 


CHAPTER    II. 

THE    UTILITY    OF    MATHEMATICS    REGARDED    AS    A    MEANS    OF    ACQUIRING 
KNOWLEDGE BACONIAN    PHILOSOPHY. 

Mathematics:      §  331.   In  the  preceding  chapter,  we  consid 
ered  the  effects  of  mathematical  studies  on  the 
mind,  merely  as  a  means  of  discipline  and  train- 
How  consid-  ing.     We  regarded  the  study  in  a  single  point 

ered  hereto 
fore:       oi  view,  viz.   as   the    drill-master   of  the    intel 
lectual   faculties  —  the    power   best   adapted   to 
bring  them  all  into  order — to  impart  strength, 
and   to   give    to    them    organization.       In    the 
HOW  now    present  chapter  we  shall  consider  the  study  un- 
red'   der  a   more  enlarged  aspect — as   furnishing   to 
man  the  keys   of  hidden   and  precious   knowl 
edge,    and  as  opening   to   his   mind  the   whole 
volume  of  nature. 


Material         §  332.  The  material  universe,  which  is  spread 

Universe. 

out  before  us,  is  the  first  object  of  our  rational 


CHAP.  II.]  MATERIAL     UNIVERSE.  309 

regards.    Material  things  are  the  first  with  which 

we  have  to  do.     The  child  plays  with  his  toys  Elements  of 

in   the   nursery,   paddles    in   the   limpid   water, 

twirls   his   top,    and   strikes  with   the   hammer. 

At  a  maturer  age  a  higher  class  of  ideas  are 

embraced.     The  earth  is  surveyed,  teeming  with 

its   products,    and   filled  with   life.     Man   looks 

around  him  with  wondering  and  delighted  eyes,   obtained  by 

mi  -.I-  -i  i       observation. 

Ihe  earth  he  stands  upon  appears  to  be  made 
of  firm  soil  and  liquid  waters.  The  land  is 
broken  into  an  irregular  surface  by  abrupt  hills 
and  frowning  mountains.  The  rivers  pursue 
their  courses  through  the  valleys,  without  any  course  of 

iKiturc  z 

apparent  cause,  and  finally  seem  to  lose  them 

selves  in  their  own  expansion.      He  notes  the 

return   of  day  and   night,   at  regular  intervals, 

turns   his  eyes   to  the  starry  heavens,  and  in 

quires  how  far  those  sentinels  of  the  night  may 

be  from  the  world  they  look  down  upon.      He 

is  yet  to  learn  that  all  is  governed  by  general    Governed 

laws  imparted  by  the  fiat  of  Him  who  created     ^fwsT 

all  things  ;  that  matter,  in  all  its  forms,  is  sub 

ject  to  those  laws  ;  and  that  man  possesses  the    Man  Pos- 

capacity  to  investigate,  develop,  and  understand 


them.     It  is  of  the  essence  of  law  that  it  in-  vestigate  and 

understand 

eludes    all   possible    contingencies,    and   insures      tnem- 
implicit   obedience;    and  such  are  the  laws  of 
nature. 


310  UTILITY     OF     MATHEMATICS.  [BOOK  III. 

§  333.  To  the  man  of  chance,  nothing  is  more 
mysterious  than  the  developments  of  science. 

Uniformity:  He  does  not  see  how  so  great  a  uniformity  can 
Variety  :  jonsist  with  the  infinite  variety  which  pervades 
every  department  of  nature.  While  no  two 
individuals  of  a  species  are  exactly  alike,  the 
resemblance  and  conformity  are  so  close,  that 
the  naturalist,  from  the  examination  of  a  sin 
gle  bone,  finds  no  difficulty  in  determining  the 
species,  size,  and  structure  of  the  animal.  So, 

They  appear  also,   in  the  vegetable   and  mineral   kingdoms  : 

in  all  things. 

all  the  structures  of  growth   or  formation,    al 
though  infinitely  varied,  are  yet  conformable  to 
like  general  laws. 
science  ne-       This  wonderful  mechanism,  displayed  in  the 

cessary  to  ,, 

the  devei-    structure  of  animals,  was  but  imperfectly  under- 


stood,  until  touched  by  the  magic  wand  of  sci 
ence.  Then,  a  general  law  was  found  to  per 
vade  the  whole.  Every  bone  is  of  that  length 

What  science  and  diameter  best  adapted  to  its  use  ;  every 
muscle  is  inserted  at  the  right  point,  and  works 
about  the  right  centre  ;  the  feathers  of  every 
bird  are  shaped  in  the  right  form,  and  the  curves 
in  which  they  cleave  the  air  are  best  adapted 

what  may   to  velocity.     It  is  demonstrable,   that  in  every 

be  demon 

strated.  case,  and  in  all  the  variety  of  forms  in  which 
forces  are  applied,  either  to  increase  power  or 
gain  velocity,  the  very  best  means  have  been 


CHAP.   II.]  PHILOSOPHY     OF     BACON.  311 

adopted  to  produce  the  desired  result.     And  why  why  it  is  so. 
should  it  not   be  so,   since  they  are  employed 
by  the  all- wise  Architect  ? 

§  334.  It  is  in  the  investigations  of  the  laws  Applications 
of  nature  that  mathematics  finds  its  widest  Mathematics, 
range  and  its  most  striking  applications. 

Experience,  aided  by  observation  and  enlight 
ened  by  experiment,  is  the  recognised  fountain      Bacon's 
of  all  knowledge  of  nature.     On  this  foundation 
Bacon  rested  his  Philosophy.     He  saw  that  the 
Deductive    process    of   Aristotle,  in  which   the 
conclusions  do  not  reach  beyond  the  premises,    Aristotle's: 
was  not  progressive.     It  might,  indeed,  improve 
the   reasoning  powers,  cultivate  habits  of  nice 
discrimination,    and   give    great    proficiency   in 
verbal  dialectics  ;  but  the  basis  was  too  narrow 
for   that   expansive    philosophy,  which    was    to    its  defects, 
unfold  and   harmonize  all   the   laws   of  nature. 
Hence,   he  suggested  a  careful  examination  of  what  Bacon 
nature  in  every  department,  and  laid  the  foun-    ! 
dations  of  a  new  philosophy.     Nature  was   to 
be  interrogated  by  experiment,  observation  .was 
to  note  the  results,  and  gather  the  facts  into  the 
storehouse  of  knowledge.      Facts,  so  obtained,  The  means  to 
were  subjected   to  analysis   and   collation,    and 
general  laws  inferred  from  such  classification  by 


312  UTILITY     OF     MATHEMATICS.  [iJOOK   III. 


Bacon's     a   reasoning  process  called   Induction.     Hence, 
inductive.    the  system  of  Bacon  is  said  to  be  Inductive. 


§  335.  This  new  philosophy  gave   a  startling 
impulse  to   the   human   mind.     Its   subject  was 
Nature — material  and  immaterial ;  its  object,  the 
discovery    and   analysis    of  those   general  laws 
what  it  did.  which  pervade,  regulate,  and  impart  uniformity 
to  all  things  ;   its  processes,  experience,  experi 
ment,  and  observation  for  the  ascertainment  of 
its  nature,    facts  •  analysis  and  comparison  for  their  classifi 
cation  ;  and  reasoning,  for  the  establishment  of 
what  aided  general  laws. .   But  the  work  would  have  been 
incomplete  without  the  aid  of  deductive  science. 
General  laws  deduced  from  many  separate  cases, 
what  it     by  Induction,  needed  additional  proof;  for,  they 

needed 

might  have  been  inferred  from  resemblances  too 
slight,  or  coincidences  too  few.  Mathematical 
science  affords  such  proofs. 


Thetruthsof      §336.  Regarding  general  laws,  established  by 
lon:    Induction,  as  fundamental  truths,  expressing  these 
by  means  of  the  analytical  formulas,  and  then 
operating  on  these  formulas  by  the  known  pro- 
How  verified  cesses  of  mathematical  science,  we  are  enabled, 

by  Analysis.  . 

not  only  to  verify  the  truths  of  induction,  but 
often  to  establish  new  truths,  which  were  hidden 
from  experiment  and  observation.  As  the  in- 


CHAP.   II.  1  EXPERIMENTAL     SCIENCE.  313 

ductive  process  may  involve  error,  while  the 
deductive  cannot,  there  are  weighty  scientific 
reasons,  for  giving  to  every  science  as  much 
of  the  character  of  a  Deductive  Science  as  pos 
sible.  Every  science,  therefore,  should  be  con-  Asfarns 
structed  with  the  fewest  and  simplest  possible 
inductions.  These  should  be  made  the  basis  of 

made  Deduc- 

deductive  processes,  by  which  every  truth,  how-       tive- 
ever   complex,    should   be    proved,   even   if  we 
chose  to  verify  the  same  by  induction,  based  on 
specific  experiments. 


§  337.    Every  branch  of  Natural   Philosophy 


Natural 


.    .       ,,  .  losophy  was 

was  originally  experimental;  each  generaliza-  expenmen- 
tion  rested  on  a  special  induction,  and  was  de 
rived  from  its  own  distinct  set  of  observations 
and  experiments.  From  being  sciences  of  pure 
experiment,  as  the  phrase  is,  or,  to  speak  more 
correctly,  sciences  in  which  the  reasonings  con-  is  now 

,,  deductive. 

sist  of  no  more  than  one  step,  and  that  a  step 

of   induction  ;  all  these  sciences   have   become, 

to  some  extent,  and  some  of  them  in  nearly  their 

whole  extent,  sciences  of  pure  reasoning  :  thus, 

multitudes  of  truths,   already  known  by  induc 

tion,  from  as  many  different  sets  of  experiments, 

have  come  to  be  exhibited  as  deductions,  or  co-    Matiu-mati- 

rollaries  from  inductive  propositions  of  a  simpler      cal  or 

and  more  universal  character.     Thus,  mechan- 


l 


314  UTILITY     OF     MATHEMATICS.  [fiOOK   III. 

Deductive  ics,  hydrostatics,  optics,  and  acoustics,  have 
successively  been  rendered  mathematical ;  and 
astronomy  was  brought  by  Newton  within  the 
laws  of  general  mechanics. 

Their  advan-  The  substitution  of  this  circuitous  mode  of 
proceeding  for  a  process  apparently  much  easier 
and  more  natural,  is  held,  and  justly  too,  to  be 
the  greatest  triumph  in  the  investigation  of  nature. 

They  rest  on  But,  it  is  necessary  to  remark,  that  although,  by 

Inductions. 

this  progressive  transformation,  all  sciences  tend 
to  become  more  and  more  deductive,  they  are 
not,  therefore,  the  less  inductive ;  for,  every  step 
in  the  deduction  rests  upon  an  antecedent  in- 
sciencesde-  duction.  The  opposition  is,  perhaps,  not  so 

ductive  or  ex 
perimental,   much  between  the  terms  Deductive  and  Induc 
tive  as  between  Deductive  and  Experimental. 


^-  sc^ence  is  experimental,   in  propor- 
tai science:   tjon  as  every  new  case,  which  presents  any  pe 
culiar  features,  stands  in  need  of  a  new  set  of 
observations    and   experiments,   and   a  fresh  in 
duction.     It  is  deductive,  in  proportion  as  it  can 
whende-    ^raw    conclusions,    respecting    cases    of  a   new 
kiiH^  by  processes  which  bring  these  cases  un 
der    old    inductions,    or    show  them    to   possess 
known  marks  of  certain  attributes. 

§  339.  We  can  now,  therefore,  perceive,  what 


CHAP.  II.]  DEDUCTIVE     SCIENCES.  315 


is  the  generic  distinction  between  sciences  that    Difference 
can  be  made  deductive  and  those  which  must, 


as   yet,    remain   experimental.      The   difference 

consists   in   our   having  been   able,   or  not  yet 

able,  to   draw  from  first   inductions   as  from  a 

general  law,  a  series  of  connected  and  depend 

ent  truths.      When  this  can    be   done,    the  de 

ductive   process    can   be    applied,  and   the    sci 

ence  becomes  deductive.     For  example  :    when   Deductive. 

Newton,  by  observing  and  comparing  the  mo 

tions  of  several  of  the  heavenly  bodies,  discov 

ered  that  all  the  motions,   whether  regular   or    Example. 

apparently  anomalous,  of  all  the  observed  bodies 

of  the  Solar  System,  conformed  to  the  law  of 

moving  around  a  common  centre,  urged   by  a 

centripetal  force,  varying  directly  as  the  mass, 

and  inversely  as  the  square  of  the  distance  from 

the  centre,  he  inferred  the  existence  of  such  a  whatNew- 

lawfor  all  the  bodies  of  the  system,  and  then  de-  ton  inferred: 

monstrated,  by  the  aid  of  mathematics,   that  no 

other  law  could  produce  the  motions.     This  is     what  he 

the  greatest  example  which  has  yet  occurred  of     Proved- 

the  transformation,   at  one  stroke,  of  a  science 

which  was  in  a  great  degree  purely  experimen 

tal,  into  a  deductive  science. 

§  340.  How  far  the  study  of  mathematics  pre-     study  of 
pares    the    mind   for   such    contemplations    and  mathematic3: 


316  UTILITY     OF     MATHEMATICS.  [BOOK  III. 


prepares  the  such  knowledge,  is  well  set  forth  by  an  old  wri 
ter,  himself  a  distinguished  mathematician.     He 

says  : 

Dr.  Barrow's      it  The,  steps  are  guided  by  no  lamp  more  clear- 
opinion. 

ly  through  the  dark  mazes  of  nature,  by  no  thread 

more  surely  through  the  infinite  turnings  of  the 

labyrinth  of  philosophy  ;  nor  lastly,  is  the  bottom 

of  truth  sounded  more  happily  by  any  other  line. 

HOW       I  wil}  not  mention  with  how  plentiful  a  stock 

mathematics 

furnish  the  of  knowledge  the  mind  is  furnished  from  these  ; 
with  what  wholesome  food  it  is  nourished,  and 
what  sincere  pleasure  it  enjoys.  But  if  I  speak 
further,  I  shall  neither  be  the  only  person  nor 
the  first,  who  affirms  it,  that  while  the  mind  is 

Abstract     abstracted,  and  elevated   from  sensible    matter, 

and  elevate       ... 

it:  distinctly  views  pure  torms,  conceives  the  beau 
ty  of  ideas,  and  investigates  the  harmony  of  pro 
portions,  the  manners  themselves  are  sensibly 
corrected  and  improved,  the  affections  composed 
and  rectified,  the  fancy  calmed  and  settled,  and 
the  understanding  raised  and  excited  to  more 
confirmed  by  divine  contemplations  :  all  of  which  I  might,  de- 

philosophers. 

fend  by  the   authority  and  confirm  by  the  suf 
frages  of  the  greatest  philosophers."* 

§  341.  Sir  John  Herschel,  in  his  Introduction 
*  Dr.  Barrow. 


CHAP.   II.]    ASTRONOMY    WITHOUT     MATHEMATICS.     317 

to  his   admirable  Treatise  on  Astronomy,  very     opinions, 
justly  remarks,  that, 

"  Admission  to  its  sanctuary  [the  science  of  Mathemat 
ical  science. 

Astronomy],  and  to  the  privileges  and  feelings   indispensa- 

~  •  -i  -i  •        i  -i  We  to  a 

of  a  votary,  is  only  to  be  gained  by  one  means —  knowledge  of 
sound  and  sufficient  knowledge  of  mathematics, 
the  great  instrument  of  all  exact  inquiry,  with 
out  which  no  man  can  ever  make  such  advances 
in  this  or  any  other  of  the  higher  departments 
of  science  as  can  entitle  him  to  form  an  inde 
pendent  opinion  on  any  subject  of  discussion 
within  their  range. 

"It  is  not  without  an  effort  that  those  who     informa- 

tion  cannot 

possess  this  knowledge  can  communicate  on  be  given 
such  subjects  with  those  who  do  not,  and  adapt 
their  language  and  their  illustrations  to  the  ne- 
cessities  of  such  an  intercourse.  Propositions 
which  to  the  one  are  almost  identical,  are  the 
orems  of  import  and  difficulty  to  the  other ;  nor 
is  their  evidence  presented  in  the  same  way  to 
the  mind  of  each.  In  treating  such  proposi-  Except 

i  i         •  ,  -,    ,  by  very  cum- 

tions,  under  such  circumstances,  the  appeal  has  brousmetn- 
to  be  made,  not  to  the  pure  and  abstract  reason, 
but  to  the  sense  of  analogy — to  practice  and 
experience  :  principles  and  modes  of  action  have 
to  be  established,  not  by  direct  argument  from 
acknowledged  axioms,  but  by  continually  refer 
ring  to  the  sources  from  which  the  axioms  them-  Reasons: 


18  UTILITY     OF      MATHEMATICS.  [BOOK  III. 


Must  begin   selves  have  been  drawn,  viz.  examples;  that  is  to 
piesteie-    Sa7>  ^7  bringing  forward  and  dwelling  on  simple 
"ts:     and  familiar  instances  in  which  the  same  prin 
ciples  and  the  same  or  similar  modes  of  action 
take  place ;  thus  erecting,   as  it  were,   in  each 
particular  case,   a  separate  induction,  and  con 
structing  at  each  step  a  little  body  of  science  to 
illustration   meet  its  exigencies.     The  difference  is  that  of 

of  the  differ 
ence  be-     pioneering  a  road  through  an  untraversed  coun- 

stmctionby  ^T'  anc^  advancing  at  ease   along  a  broad  and 
scientific  and  beaten  highway  i  that  is  to  say,  if  we  are  deter- 

unscientific  J  J 

methods,     mined  to  make  ourselves  distinctly  understood, 

and  will  appeal  to  reason  at  all/'     Again : 
Mathematics       "  A  certain  moderate  degree  of  acquaintance 
^k  a^stract  science  is  highly  desirable  to  every 
one  who  would  make  any  considerable  progress 
in  physics.     As  the  universe  exists  in  time  and 
place  ;  and  as  motion,  velocity,  quantity,  num 
ber,  and  order,  are  main  elements  of  our  knowl 
edge  of  external  things    and  their  changes,   an 
acquaintance    with    these,    abstractedly    consid- 
is  so  ered  (that  is  to  say,  independent  of  any  consid 
eration  of  particular  things    moved,    measured, 
counted,  or  arranged),  must  evidently  be  a  use 
ful  preparation  for  the  more  complex  study  of 
nature/'* 


*  Sir  John  Herschel  on  the  study  of  Natural  Philosophy. 

II 


CHAP.   II.]  ASTRONOMY.  310 

§  342.  If  we  consider  the  department  of  chem-  Necessary  in 

chemistry. 

istry, — which  analyzes  matter,  examines  the  ele 
ments  of  which  it  is  composed,  develops  the  laws 
which  unite  these  elements,  and  also  the  agencies 
which  will  separate  and  reunite  them, — we  shall 
find  that  no  intelligent  and  philosophical  analysis 
can  be  made  without  the  aid  of  mathematics. 

§  343.    The  mechanism  of  the  physical  uni-    Laws  long 

unknown. 

verse,  and  the  laws  which  govern  and  regulate 
its  motions,  were  long  unknown.     As  late  as  the 
17th  century,   Galileo  was  imprisoned  for  pro-      Galileo, 
mulgating  the  theory  that  the  earth  revolves  on 
its  axis;  and  to  escape  the  fury  of  persecution,  His  theory. 
renounced  the  deductions  of  science.     Now,  ev 
ery  student  of  a  college,  and  every  ambitious  boy  NOW  known 
of  the  academy,  may,  by  the  aid  of  his  Algebra 
and  Geometry,  demonstrate   the   existence    and 
operation  of  those   general  laws  which  enable     By  what 

means  de- 

him  to  trace  with  certainty  the   path  and   mo-   monstrated. 
tions  of  every  body  which  circles  the  heavens. 

§  344.  What  knowledge  is  more  precious,  or  Vaiue 
more  elevating  to  the  mind,  than  that  which 
assures  us  that  the  solar  system,  of  which  the 
sun  is  the  centre,  and  our  earth  one  of  the 
smaller  bodies,  is  governed  by  the  general  law 
of  gravitation ;  that  is,  that  each  body  is  re 
tained  in  its  orbit  by  attracting,  and  being  at- 


L_ 


320  UTILITY     OF     MATHEMATICS.  [BOOK    III. 

what      tracted  by,  all  the  others  ?    This  power  of  attrac- 

it  teaches.  ,  ,  .    .  .        . 

tion,  by  which  matter  operates  on  matter,  is  the 

great  governing  principle  of  the  material  world. 

The   motion  of  each  body  in  the  heavens  de- 

The  things   pends    on    the   forces    of  attraction    of    all    the 

not  easy.  ,  •  i      r- 

others  ;  hence,  to  estimate  such  lorces — varying 
as  they  do  with  the  quantity  of  matter  in  each 
body,  and  inversely  as  the  squares  of  their  dis 
tances  apart — is  no  easy  problem  ;  yet  analy- 
Anaiysis:  sis  has  solved  it,  and  with  such  certainty,  that 
the  exact  spot  in  the  heavens  may  be  marked 
at  which  each  body  will  appear  at  the  expiration 

What  it  has  of  any  definite  period  of  time.     Indeed,  a  tele- 
done: 

scope   may  be   so   arranged,  that  at  the   end  of 

HOW  a      that   time  either   one    of    the   heavenly   bodies 
be  verified   would  present  itself  to  the  field  of  view ;    and 
by^ten"    if  the  instrument  could  remain  fixed,  though  the 
time  were    a    thousand  years,    the  precise  mo 
ment  would  discover  the  planet  to  the  eye  of  the 
observer,  and  thus  attest  the  certainty  of  science. 

§  345.  But   analysis  has   done  yet   more.     Tt 

has  not  only  measured  the  attractive  power  of 

Analysis     each  of  the  heavenly  bodies  ;  determined   their 

determines 

balancing     distances  from  a  common  point  and  from  each 

forces. 

other;  ascertained  their  specific  gravities  and 
traced  their  orbits  through  the  heavens ;  but 
has  also  discovered  the  existence  of  balancing 


CHAP.   II.]      STABILITY     OF     THE     UNIVERSE.  3J1 

and   conservative   forces,   evincing  the   highest  Evidence  of 
evidence  of  contrivance  and  design. 


§  346.  A  superficial  view  of  the  architecture 
of  the  heavens  might  inspire  a  doubt  of  the  sta 
bility  of  the  entire  system.  The  mutual  action 
of  the  bodies  on  each  other  produces  what  is 
called  an  irregularity  in  their  motions.  The 
earth,  for  example,  in  her  annual  course  around 
the  sun,  is  affected  by  the  attraction  of  the 
moon  and  of  all  the  planets  which  compose  the 
solar  system ;  and  these  attracting  forces  appear 
to  give  an  irregularity  to  her  motions.  The 
moon  in  her  revolutions  around  the  earth  is  also 
influenced  by  the  attraction  of  the  sun,  the  earth, 
and  of  all  the  other  planets,  and  yields  to  each  a 
motion  exactly  proportionate  to  the  force  exert 
ed  ;  and  the  same  is  equally  true  of  all  the  bodies 
which  belong  to  the  system.  It  was  reserved 
for  analysis  to  demonstrate  that  every  supposed 
irregularity  of  motion  is  but  the  consequence  of 
a  general  law ;  that  every  change  is  constancy, 
and  every  diversity  uniformity.  Thus,  mathe 
matical  science  assures  us  that  our  system  has 
not  been  abandoned  to  blind  chance,  but  that 
a  superintending  Providence  is  ever  exerted 
through  those  general  laws,  which  are  so  minute 

as  to  govern  the  motions  of  the  feather  as  it  is 

21 


Architecture 
of  the  heav 
ens  shows 
permanency. 


Example  of 
the  earth : 


Of  the  moon. 


Of  the  other 
planets. 


Mathematics 
proves  the 

permanency 
of  the  sys 
tem. 


822  UTILITY     OF     MATHEMATICS.  [BOOK  III. 


so 


Generality  of  wafted  along  on  the  passing  breeze,  and  yet  s 
omnipotent  as  to  preserve  the  stability  of  worlds 

§  347.    But   analysis   goes  yet    another   step. 

That  class  of  wandering  bodies,  known  to  us  by 

comets:     the  name  of  comets,  although  apparently  escaped 

from  their  own  spheres,  and  straying  heedlessly 

what      through  illimitable  space,  have  yet  been  pursued 

mathematics  J 

proves  in  re-  by  the  telescope  of  the  observer  until  sufficient 

gard  to  them. 

data  have  been  obtained  to  apply  the  process 
of  analysis.  This  done,  a  few  lines  written  upon 
paper  indicate  the  precise  times  of  their  reap- 

Rosuits  stri-  pearance.  These  results,  when  first  obtained, 
were  so  striking,  and  apparently  so  far  beyond 
the  reach  of  science  itself,  as  almost  to  need 

Verification,  the  verification  of  experience.  At  the  appointed 
times,  however,  the  comets  reappear,  and  sci 
ence  is  thus  verified  by  observation. 


Nature          §  348.  The  great  temple  of  nature  is  only  to 

cannot  be  in-  ... 

vestigated    be  opened  by  the  keys  of  mathematical  science. 
may  PernaPs  reach  the  vestibule,  and  gaze 


with  wonder  on  its  gorgeous  exterior  and  its 
exact  proportions,  but  we  cannot  open  the  por- 
niustration.  tal  and  explore  the  apartments  unless  we  use 
the  appointed  means.  Those  means  are  the 
exact  sciences,  which  can  only  be  acquired  by 
discipline  and  severe  mental  labor. 


CHAP.   II.]  RESULTS      OF     SCIENCE.  323 

The  precious  metals  are  not  scattered  pro-  science: 
fusely  over  the  surface  of  the  earth  ;  they  are, 
for  wise  purposes,  buried  in  its  bosom,  and  can 
be  disinterred  only  by  toil  and  labor.  So  with 
science  :  it  comes  not  by  inspiration  ;  it  is  not 
borne  to  us  on  the  wings  of  the  wind  ;  it  can  only  to  be 

acquired  by 

neither  be  extorted  by  power,  nor  purchased  by      8ludy: 
wealth  ;  but  is  the  sure  reward  of  diligent  and 
assiduous  labor.     Is  it  worth  that  labor  ?     What   Tt  fc  worth 

study. 

is  it  not  worth  ?     It  has  perforated  the  earth, 


and  she  has  yielded  up  her  treasures  ;    it   has 

it  has  done 

guided  in  safety  the  bark  of  commerce  over  dis-  for  the  wanta 

.  of  man  : 

tant  oceans,  and  brought  to  civilized  man  the 
treasures  and  choicest  products  of  the  remotest 
climes.  It  has  scaled  the  heavens,  and  searched 
out  the  hidden  laws  which  regulate  and  govern 
the  material  universe  ;  it  has  travelled  from 
planet  to  planet,  measuring  their  magnitudes,  sur 
veying  their  surfaces,  determining  their  days  and 
nights,  and  the  lengths  of  their  seasons.  It  has 
also  pushed  its  inquiries  into  regions  of  space,  what 

•      i  t  f.      i          it  nu9  done 

where  it  was   supposed  that   the   mind   of   the   to  make  us 


Omnipotent  never  yet  had  energized,  and  there  ^" 
located   unknown   worlds  —  calculating  their  di-       verse- 
ameters,  and  their  times  of  revolution. 

§  349.    Mathematical   science   is   a   magnetic       How 
telegraph,  which  conducts  the   mind  from   orb  mathematlcs 


324  UTILITY     OF     MATHEMATICS.  [BOOK  III. 

aid  the      to  orb  through  the  entire  regions  of  measured 

mind  in  its 

inquiriea:  space.  It  enables  us  to  weigh,  in  the  balance 
of  universal  gravitation,  the  most  distant  planet 
of  the  heavens,  to  measure  its  diameter,  to  de 
termine  its  times  of  revolution  about  a  common 
centre  and  about  its  own  axis,  and  to  claim  it 
as  a  part  of  our  own  system. 

HOW  they        In  these  far  Teachings  of  the  mind,  the  im- 

enlarge  it: 

agination  has  full  scope  for  its  highest  exercise. 
It  is  not  led  astray  by  the  false  ideal  and  fed  by 
illusive  visions,  which  sometimes  tempt  reason 
from  her  throne,  but  is  ever  guided  by  the  de- 
May  be  ductions  of  science ;  and  its  ideal  and  the  real 
are  united  by  the  fixed  laws  of  eternal  truth. 

Mind  §  350.  There  is  that  within  us  which  delights 

certainty.  m  certainty.  The  mists  of  doubt  obscure  the 
mental,  as  the  mists  of  the  morning  do  the  phys 
ical  vision.  We  love  to  look  at  nature  through 
a  medium  perfectly  transparent,  and  to  see  every 
object  in  its  exact  proportions.  The  science  of 

why       mathematics  is  that  medium  through  which  the 

mathematics 

afford  it.  mind  may  view,  and  thence  understand  all  the 
parts  of  the  physical  universe.  It  makes  man 
ifest  all  its  laws,  discovers  its  wonderful  harmo 
nies,  and  displays  the  wisdom  and  omnipotence 
of  the  Creator. 


CHAP.  HI.]  "PRACTICAL."  325 


CHAPTER    III. 


THE   UTILITY   OF   MATHEMATICS    CONSIDERED   AS   FURNISHING   THOSE   RULES   OF 
ART   WHICH   MAKE    KNOWLEDGE   PRACTICALLY   EFFECTIVE. 


§  351.  THERE  is  perhaps  no  word  in  the  Eng-    practical: 
lish  language   less   understood  than   PRACTICAL.      Little 

-r,  .  ,     ,  ,  .      understood. 

By  many  it  is  regarded  as  opposed  to  theoreti 
cal.     It  has  become  a  pert  question  of  our  day,   its  popular 
"  Whether  such  a  branch  of  knowledge  is  prac 
tical  ?"     "  If  any  practical  good  arises  from  pur-    Questions 

relfi tin0*  to 

suing  such  a  study?"     "If  it  be  not  full  time  gtudiesand 
that  old  tomes  be  permitted  to  remain  untouched 
in  the  alcoves  of  the  library,  and  the  minds  of 
the  young  fed  with  the  more  stimulating  food  of 
modern  progress  ?" 


§  352.  Such  inquiries  are  not  to  be  answered  inquiries: 
by  a  taunt.     They  must  be  met  as  grave  ques-  H°^  to  be 
tions,  and  considered  and  discussed  with  calm 
ness.     They  have  possession  of  the  public  mind ;  ^^ 

influence. 

they  affect  the  foundations  of  education ;    they 


326  UTILITY     OF     MATHEMATICS.  [BOOK  III. 


Their      influence  and  direct  the  first  steps  ;  they  control 
the  very  elements  from  whici 
systems  of  public  instruction. 


the  very  elements  from  which  must   spring  the 


Practical:        §353.  The  term  "practical,"  in  its  common 
common    acceptation,  that  is,  in  the  sense  in  which  it  is 
often  used,  refers  to   the   acquisition   of  useful 
knowledge  by  a  short  process.     It  implies  a  sub- 
whatu     stitution  of  natural  sagacity  and  "mother  wit" 
lmphe8'     for  the  results  of  hard  study  and  laborious  effort. 
It  implies  the  use  of  knowledge  before  its  acqui 
sition  ;    the  substitution  of  the  results  of  mere 
experiment  for  the  deductions  of  science,   and 
the  placing  of  empiricism  above  philosophy. 

in  this  sense,  §  354.  In  this  view,  the  practical  is  adverse 
to  soun^  learning,  and  directly  opposed  to  real 
progress.  If  adopted,  as  a  basis  of  national  edu 
cation,  it  would  shackle  the  mind  with  the  iron 
fetters  of  mere  routine,  and  chain  it  down  to 
the  drudgery  of  unimproving  labor.  Under 
such  a  system,  the  people  would  become  imita- 
tors  and  rule-men.  Great  and  original  principles 
would  be  lost  sight  of,  and  the  spirit  of  inves 
tigation  and  inquiry  would  find  no  field  for  its 
legitimate  exercise. 

Eight  But   give  to    "practical"  its   true   and  right 

signification,    and   it   becomes    a    word   of    the 


CHAP.   III.]  ILLUSTRATED.  327 

choicest  import.     In  its  right  sense,  it  is  the  best   Best  means 
means  of  making  the  true  ideal  the  actual ;  that  ^^5^ 
is,  the  best  means   of  carrying  into  the  business 
and  practical  affairs  of  life  the  conceptions  and 
deductions  of  science.      All  that  is  truly  great 
in  the  practical,  is  but  the  actual  of  an  antece 
dent  ideal. 

§  355.  It  is  under  this  view  that  we  now  pro-    Mathemati 
cal  science : 

pose  to  consider  the  practical  advantages  of 
mathematical  science.  In  the  two  preceding 
chapters  we  have  pointed  out  its  value  as  a 
means  of  mental  development,  and  as  affording 
facilities  for  the  acquisition  of  knowledge.  We 
shall  now  show  how  intimately  it  is  blended  its  practical 
with  the  every-day  affairs  of  life,  and  point  out 
some  of  the  agencies  which  it  exerts  in  giving 
practical  development  to  the  conceptions  of  the 
mind. 

§  356.  We  begin  with  Arithmetic,  as  this  Arithmetic 
branch  of  mathematics  enters  more  or  less  into 
all  the  others.  And  what  shall  we  say  of  its 
practical  utility  ?  It  is  at  once  an  evidence  and 
element  of  civilization.  By  its  aid  the  child  in 
the  nursery  numbers  his  toys,  the  housewife 
keeps  her  daily  accounts,  and  the  merchant  sums 
up  his  daily  business.  The  ten  little  characters, 


328  UTILITY     OF     MATHEMATICS.  [BOOK  III. 

which  we  call  figures,  thus  perform  a  very  im- 

what  figures  portant  part  in  human  affairs.  They  are  sleepless 
sentinels  watching  over  all  the  transactions  of 
trade  and  commerce,  and  making  known  their 
final  results.  They  superintend  the  entire  busi- 

Their  value,  ness  affairs  of  the  world.  Their  daily  records 
exhibit  the  results  on  the  stock  exchange,  and 
of  enterprises  reaching  over  distant  seas.  The 

used  by  the  mechanic  and  artisan  express  the  final  results  of 
all  their  calculations  in  figures.  The  dimensions 

in  bunding,  of  buildings,  their  length,  breadth,  and  height,  as 
well  as  the  proportions  of  their  several  parts,  are 
all  expressed  by  figures  before  the  foundation 

Aid  science,  stones  are  laid ;  and  indeed,  all  the  results  of 
science  are  reduced  to  figures  before  they  can 
be  made  available  in  practice. 

§  357.  The  rules  and  practice  of  all  the  me 
chanic  arts  are  but  applications  of  mathematical 
Mathematics  science.     The  mason  computes  the  quantity  of 

useful  in  the 

mechanic  his  materials  by  the  principles  of  Geometry  and 
the  rules  of  Arithmetic.  The  carpenter  frames 
his  building,  and  adjusts  all  its  parts,  each  to 
the  others,  by  the  rules  of  practical  Geometry. 

Examples.  The  millwright  computes  the  pressure  of  the 
water,  and  adjusts  the  driving  to  the  driven 
wheel,  by  rules  evolved  from  the  formulas  of 
analvsis. 


CHAP.   III.]  ILLUSTRATED.  329 

§  358.  Workshops  and  factories  afford  marked  workshops 
illustrations  of  the  utility  and  value  of  practical  exhiWtap. 
science.  Here  the  most  difficult  problems  are  potions  of 

science. 

resolved,  and  the  power  of  mind  over  matter 
exhibited  in  the  most  striking  light.  To  the 
uninstructed  eye  of  a  casual  observer,  confusion 

appears  to  reign  triumphant.     But  all  the  parts     Parts  ad 
justed  on  a 
of  that  complicated  machinery   are  adjusted  to  general  plan. 

each  other,  and  were  indeed  so  arranged,  and 

according   to   a   general   plan,   before   a   single 

wheel  was  formed  by  the  hand  of  the  forger. 

The  power  necessary  to  do  the  entire  work  was      Power 

first   carefully  calculated,    arid   then   distributed       and 

throughout  the  ramifications  of  the  machinery. 

Each  part  was  so  arranged  as  to  fulfil  its  office. 

Every  circumference,   and   band,  and  cog,  has 

its   specific   duty   assigned   it.      The   parts   are    Parts  fit  in 

their  proper 

made  at  different  places,  after  patterns  formed      places, 
by  the  rules  of  science,  and  when  brought  to 
gether,  fit  exactly.     They  are  but  formed  parts 
of  an  entire  whole,  over  which,   at  the  source 
of  power,  an  ingenious  contrivance,  called  the 
Governor,  presides.     His  function  is  to  regulate   Governor: 
the  force  which  shall  drive  the  whole  according 
to  a  uniform  speed.     He  is   so  intelligent,  and 
of  such  delicate  sensibility,  that  on  the  slightest  its  functions. 
increase  of  velocity,  he  diminishes  the  force,  and 
adds   additional   power  the   moment   the   speed 


330  UTILITY     OF     MATHEMATICS.  [BOOK  III. 


AII  is  but    slackens.     All  this  is  the  result  of  mathematical 

the  result  of  .  TTTI 

science,  calculation.  When  the  curious  shall  visit  these 
exhibitions  of  ingenuity  and  skill,  let  them  not 
suppose  that  they  are  the  results  of  chance  and 
experiment.  They  are  the  embodiments,  by  in 
telligent  labor,  of  the  most  difficult  investigations 
of  mathematical  science. 

§  359.  Another  striking  example  of  the  appli 
cation   of  the  principles  of  science  is  found  in 
steamship:  the  steamship. 

in  the  first  place,   the  formation  of  her  hull, 
HOW  the  huii  so  as  to  divide  the  waters  with  the  least  resist- 
ied'    ance,  and  at  the  same   time  receive  from  them 
the  greatest  pressure  as  they  close  behind  her, 
Her  masts:    is    not  an   easy  problem.      Her   masts    are    all 
How       to  be  set  at  the  proper  angle,  and  her  sails  so 
adjusted.     acijuste(j  as  to  gain  a  maximum  force.     But  the 
complication    of    her    machinery,    unless    seen 
through  the  medium  of  science,  baffles  investi 
gation,  and  exhibits    a  startling    miracle.     The 
burning  furnace,  the  immense  boilers,  the  mass- 
Machinery:  iyc   cylinders,    the   huge   levers,   the   pipes,   the 
lifting  and   closing  valves,   and    all  the    nicely- 
adjusted  apparatus,  appear  too   intricate  to   be 
comprehended  by  the  mind  at  a  single  glance. 
The  whole    Yet  in  all  this  complication — in  all  this  variety 
of  principle   and  workmanship,   science  has  ex- 


CHAP.   III.]  ILLUSTRATED.  331 


erted  its  power.     There  is  not  a  cylinder,  whose  according  to 

the  principles 

dimensions   were   not   measured  —  not    a  lever,    Of science: 
whose  power  was  not  calculated — nor  a  valve, 
which  does  not  open  and  shut  at  the  appointed 
moment.     There  is  not,  in  all  this  structure,  a     From  a 

.  general  plan. 

bolt,  or  screwr,  or  rod,  which  was  not  provided 
for  before  the  great  shaft  was  forged,  and  which 
does  not  bear  to  that  shaft  its  proper  proportion. 
And  when  the  workmanship  is  put  to  the  test,  B>- 

what  means 

and  the  power  of  steam  is  urging  the  vessel  on   liavigated: 
her  distant  voyage,  science  alone  can  direct  her 
way. 

In  the  captain's  cabin  are  carefully  laid  away, 
for  daily  use,  maps  and  charts  of  the  port  which  Her  charts: 
he  leaves,  of  the  ocean  he  traverses,  and  of  the 
coasts  and  harbors  to  which  he  directs  his  way. 
On  these  are  marked  the  results  of  much  scien-       Their 
tific  labor.     The  shoals,  the  channels,  the  points 


uses. 


of  danger  and  the  places  of  security,  are  all  in 
dicated.      Near  by,  hangs   the  barometer,   con-    Barometer: 
structed  from  the  most   abstruse   mathematical 
formulas,  to  indicate  changes   in  the  weight  of 
the   atmosphere,   and    admonish  him  of  the   ap 
proaching  tempest.     On  his  table  lie  the  sextant,     sextant: 
and  the  tables  of  Bowditch.     These  enable  him, 
by  observations  on  the  heavenly  bodies,  to  mark 
his  exact  place  on  the  chart,  and  learn  his  posi-   Their  uses. 
tion  on  the  surface  of  the  earth.     Thus,  practical 


332  UTILITY     OF     MATHEMATICS.  [BOOK  III. 


science      science,  which  shaped  the  keel  of  the  ship  to 

guides  the 

ship :  its  proper  form,  and  guided  the  hand  of  the  me 
chanic  in  every  workshop,  is,  under  Providence, 
the  means  of  conducting  her  in  safety  over  the 
ocean.  It  is,  indeed,  the  cloud  by  day  and  the 

What      pillar  of  fire   by  night.      Guiding   the   bark  of 

thusaccom-    r 

pushes,  commerce  over  trackless  waters,  it  brings  dis 
tant  lands  into  proximity,  and  into  political  and 
social  relations. 

"  We  have  before  us  an  anecdote  communi- 

Illustration. 

cated   to   us    by   a  naval   officer,*   distinguished 

for  the   extent    and  variety  of  his   attainments, 

which  shows   how  impressive  such  results  may 

Ca  t  Hairs   becom6  in  practice.     He  sailed  from  San  Bias, 

voyage.      on  ^  west  coast  of  Mexico,  and  after  a  voyage 

its  length:    of  eight  thousand  miles,  occupying  eighty-nine 

days,  arrived  off  Rio  de  Janeiro ;  having  in  this 

interval  passed  through  the  Pacific  Ocean,  round- 

and       ed  Cape  Horn,  and  crossed  the  South  Atlantic, 

incidents. 

without  making  any  land,  or  even  seeing  a  single 
sail,  with  the  exception  of  an  American  whaler 
off  Cape  Horn.  Arrived  within  a  week's  sail 
of  Rio,  he  set  seriously  about  determining,  by 
observations  lunar  observations,  the  precise  line  of  the  ship's 

taken. 

course,  and  its  situation  in  it,  at  a  determinate 
moment;    and    having   ascertained    this   within 


*  Captain  Basil  Hall. 


CHAP.  III.]  ILLUSTRATED.  333 

from  five  to  ten  miles,  ran  the  rest  of  the  way  Remarkable 
by  those  more  ready  and  compendious  methods, 
known  to  navigators,  which  can  be  safely  em 
ployed  for  short  trips  between  one  known  point 
and  another,  but  which  cannot  be  trusted  in  long      short 

methods. 

voyages,  where  the  moon  is  the  only  sure  guide. 

"  The  rest  of  the  tale,  we  are  enabled,  by  his 
kindness,  to  state  in  his  own  words  :  '  We  steered  Particulars 
towards  Rio  de  Janeiro  for  some  days  after  ta 
king  the  lunars  above  described,  and  having 
arrived  within  fifteen  or  twenty  miles  of  the  Arrival  at 
coast,  I  hove-to  at  four  in  the  morning,  till  the 
day  should  break,  and  then  bore  up  :  for  although 
it  was  very  hazy,  we  could  see  before  us  a  couple 
of  miles  or  so.  About  eight  o'clock  it  became  so 
foggy,  that  I  did  not  like  to  stand  in  further,  and 
was  just  bringing  the  ship  to  the  wind  again,  be 
fore  sending  the  people  to  breakfast,  when  it  sud 
denly  cleared  off,  and  I  had  the  satisfaction  of  Discovery  of 

Harbor. 

seeing  the  great  Sugar-Loaf  Rock,  which  stands 
on  one  side  of  the  harbor's  mouth,  so  nearly  right 
ahead  that  we  had  not  to  alter  our  course  above 
a  point  in  order  to  hit  the  entrance  of  Rio.  This 
was  the  first  land  we  had  seen  for  three  months,  First  land  in 

,    ,     .  i        ,  three 

alter  crossing  so  many  seas,  and  being  set  back-     monthg. 
wards    and   forwards   by   innumerable   currents 
and  foul  winds.'      The  effect  on  all  on   board      Effect 
might  well  be  conceived  to  have  been  electric  ; 


334  UTILITY     OF     MATHEMATICS.  [BOOK    III. 

on  the  crew,  and  it  is  needless  to  remark  how  essentially  the 
authority  of  a  commanding  officer  over  his  crew 
may  be  strengthened  by  the  occurrence  of  such 
incidents,  indicative  of  a  degree  of  knowledge 
and  consequent  power  beyond  their  reach."* 

Surveying.        §  360.    A  useful   application  of  mathematical 
science  is  found  in  the  laying  out  and  measure- 
Measure-    ment  of  land.     The  necessity  of  such  measure 
ment  of  land. 

ment,  and  of  dividing  the  surface  of  the  earth 

into  portions,  gave  rise  to  the  science  of  Geom- 
ownership:  etry.     The  ownership  of  land  could  not  be  de- 
How       termined  without  some  means  of  running  boun 

determined. 

dary  lines,  and  ascertaining  limits.  Levelling 
is  also  connected  with  this  branch  of  practical 
mathematics. 

By  the  aid  of  these  two  branches  of  practical 
science,  we  measure  and  determine  the  area  or 

contents  of  contents  of  ground ;  make  maps  of  its  surface  ; 
measure  the  heights  of  hills  and  mountains ; 
Rivers,  find  the  directions  of  rivers ;  measure  their  vol 
umes,  and  ascertain  the  rapidity  of  their  cur 
rents.  So  certain  and  exact  are  the  results,  that 
entire  countries  are  divided  into  tracts  of  con 
venient  size,  and  the  rights  of  ownership  fully 

Certainty,    secured.     The  rules  for  mapping,  and  the  con- 

*  Sir  John  Herschel,  on  the  study  of  Natural  Philosophy. 


CHAP.  III.]  ILLUSTRATED.  335 

ventional  methods  of  representing   the   surface    Mapping, 
of  ground,  the  courses  of  rivers,  and  the  heights 
of  mountains,  are  so  well  defined,  that  the  nat 
ural  features  of  a  country  may  be  all  indicated   Features  of 
on  paper.     Thus,  the  topographical  features  of 
all  the  known  parts  of  the  earth  may  be  cor-  Their  repre- 
rectly  and  vividly  impressed  on  the  mind,  by  a 
map,  drawn  according  to  the  rules  of  art,  by  the 
human  hand. 


§  361.   Our  own  age  has  been  marked  by  a    Railways, 
striking  application  of  science,  in  the  construc 
tion  of  railways.     Let  us  contemplate  for  a  mo-  The  problem 
ment  the  elements  of  the  problem  which  is  pre-    pres 
sented  in  the  enterprise  of  constructing  a  railroad 
between  two  given  points. 

In  the  first  place,  the  route  must  be  carefully  Examination 
examined  to  ascertain  its  general  practicability,      routes!" 
The  surveyor,  with  his  instruments,  then  ascer-     surveys. 
tains  all  the  levels   and  grades.     The  engineer 
examines  these  results  to  determine  whether  the  office  of  the 
power  of  steam,  in    connection  with    the   best 
combination   of  machinery,  will  enable  him   to 
overcome  the  elevations  and  descend  the  decliv 
ities  in  safety.     He  then  calculates  the  curves  calculations 
of  the   road,   the  excavations    and   fillings,   the 
cost  of  the  bridges  and  the  tunnels,  if  there  are 
any ;  and  then  adjusts  the  steam-power  to  meet 


336  UTILITY     OF     MATHEMATICS.  [fiOOK  III. 

Completion   the  conditions.     In  a  few  months  after  the  enter- 

and  use. 

prise  is  undertaken,  the  locomotive,  with  its  long 
train  of  passenger  and  freight  cars,  rushes  over 
the  tract  with  a  superhuman  power,  and  fulfils 
the  office  of  uniting  distant  places  in  commer 
cial  and  social  relations. 
The  striking  But  that  which  is  most  striking  in  all  this,  is 

f'ict 

the  fact,  that  before  a  stump  is  grubbed,  or  a 
spade  put  into  the  ground,  the  entire  plan  of  the 
work,  having  been  subjected  to  careful  analysis, 
is  fully  developed  in  all  its  parts.  The  construc- 
The  whole  tion  is  but  the  actual  of  that  perfect  ideal  which 


^bnflB,  °f  the  mind  forms  within  itself,  and  which  can 
spring  only  from  the  far-reaching  and  immuta 
ble  principles  of  abstract  science. 

§  362.  Among  the  most  useful  applications  of 

practical  science,  in  the  present  century,  is  the 

croton      introduction  of  the  Croton  water  into  the  city 

.Deduct. 


In  the  Highlands  of  the  Hudson,  about  fifty 
miles  from  the  city,  the  gushing  springs  of  the 

sources  of  mountains  indicate  the  sources  of  the  Croton 
river,  which  enters  the  Hudson  a  few  miles 
below  Peekskill.  At  a  short  distance  from  the 

Principal     mouth,  a  dam  fifty-five  feet  in  height  is  thrown 

reservoir. 

across  the  river,  creating  an  artificial  lake  for 
the  permanent  supply  of  water.     The  area  of  this 


CHAP,  in.] 


ILLUSTRATED. 


337 


Their 
heights. 


Streams 
crossed. 


lake  is  equal  to  about  four  hundred  acres.     The      its  area. 
aqueduct  commences  at  the  Croton  dam,  on  a   Aqueduct, 
line  forty  feet  above  the  level  of  the  Hudson 
river,  and   runs,  as  near  as   the  nature  of  the 
ground  will  permit,  along  the  east  bank,  till  it 
reaches   its   final    destination    in    the   reservoirs 
of  the  city.     There  are  on  the  line  sixteen  tun-   its  tunnels; 
nels,  varying  in  length  from  160  to  1,263  feet, 
making  an  aggregate  length  of  6,841  feet.     The 
heights  of  the  ridges  above  the  grade  level  of  the 
tunnels  range  from  25  to  75  feet.     Twenty-five 
streams  are  crossed  by  the  aqueduct  in  West- 
chester  county,  varying  from  12  to  70  feet  below 
the  grade  line,  and  from  25  to  83  feet  below  the 
top    covering   of  the    aqueduct.      The    Harlem  Harlem  river: 
river  is  passed  at  an  elevation  of  120  feet  above 
the  surface  of  the  water.     The  average  dimen 
sions  of  the  interior  of  the  aqueduct,  are  about 
seven  feet  in  width  and  eight  feet  in  height. 

The  width  of  the  Harlem  river,  at  the  point    its  width, 
where  the  aqueduct    crosses  it,   is  six  hundred 
and  twenty  feet,  and   the  general  plan  of  the 
bridge  is  as   follows  :    There  are  eight  arches,      Bridge : 
each  of  80  feet  span,  and  seven  smaller  arches, 
each  of  50  feet  span,  the  whole  resting  on  piers 
and  abutments.      The   length   of   the  bridge  is    its  length: 
1,450  feet.     The  height  of  the  river  piers  from 
the  lowest  foundation  is  96   feet.     The   arches 

22 


338  UTILITY     OF     MATHEMATICS.  [BOOK  III. 


its  height:  are  semi-circular,  and  the  height  from  the  low 
est  foundation  of  the  piers  to  the  top  of  the 

its  width,  parapet  is  149  feet.  The  width  across,  on  the 
top,  is  21  feet. 

To  afford  a  constant  supply  of  water  for  dis 
tribution  in  the  city  two  large  reservoirs  have 

Receiving  been  constructed,  called  the  receiving  reservoir 
and  the  distributing  reservoir.  The  surface  of 
the  receiving  reservoir,  at  the  water-line,  is  equal 

cts  extent,  to  thirty-one  acres.  It  is  divided  into  two  parts 
by  a  wall  running  east  and  west.  The  depth  of 

Dc>pth  of    water  in  the   northern  part  is  twenty  feet,  and 

water.          . 

in  the  southern  part  thirty  feet. 

Distributing       The  distributing  reservoir  is  located  on  the 
:    highest  ground  which  adjoins  the  city,   known 

its  capacity,  as  Murray  Hill.     The  capacity  of  this  reservoir 
is  equal  to  20,000,000  of  gallons,  which  is  about 
one-seventh  that  of  the  receiving  reservoir,  and 
the  depth  of  water  is  thirty- six  feet. 
Power  The  full  power  of  science  has  not  yet  been 

illustrated.  A  perfect  plan  of  this  majestic 
structure  was  arranged,  or  should  have  been, 
before  a  stone  was  shaped,  or  a  pickaxe  put  into 
the  ground.  The  complete  conception,  by  a 
single  mind,  of  its  general  plan  and  minutest 
details,  was  necessary  to  its  successful  prosecu- 

whatitac-  tjon>      jt  was  within  the  range   and  power  of 

complished. 

science  to  have  given  the  form  and  dimensions 


CHAP.  III.]  ILLUSTRATED.  339 


of  every  stone,  so  that  each  could  have  been 
shaped  at  the  quarry.  The  parts  are  so  con- 
nected  by  the  laws  of  the  geometrical  forms, 
that  the  dimensions  and  shape  of  each  stone  was 
exactly  determined  by  the  nature  of  that  portion 
of  the  structure  to  which  it  belonged. 


§  363.  We  have  presented  this  outline  of  the  view  of  the 
Croton    aqueduct    mainly   for   the    purpose   of    aqueduct: 
illustrating  the   power  and  celebrating   the   tri-  why  given. 
umphs    of    mathematical  science.      High  intel 
lect,  it  is   true,  can  alone  use  the   means  in  a 
work  so   complicated,   and  embracing  so  great 
a  variety  of  intricate  details.     But  genius,  even     Little  ao_ 
of  the  highest  order,  could  not  accomplish,  with-   ™™^f 
out  continued   trial    and   laborious   experiment,      science- 
such   an   undertaking,  unless   strengthened   and 
guided  by  the  immutable  truths  of  mathematical 
science. 

§  364.  The  examination  of  this  work  cannot      what 

.  „,,      ,  ,        .   ,  .  c    science  has 

but  fill  the  mind  with  a  proud  consciousness  of  done 
the  power  and  skill  of  man.  The  struggling 
brooks  of  the  mountains  are  collected  together  — 
accumulated  —  conducted  for  forty  miles  through 
a  subterranean  channel,  to  form  small  lakes  in 
the  vicinity  of  a  populous  city. 

From  these  sources,  by  an  unseen  process,  the 


340  UTILITY     OF     MATHEMATICS.  [BOOK   III. 


pure  water  is  carried  to  every  dwelling  in  the 

large  metropolis.      The  turning  of  a  faucet  de- 

conse-      livers  it  from  a  spring  at  the  distance  of  fifty 

quences 

which  have  miles,  as  pure  as  when  it  gushes  from  its  granite 

followed.       I  MI  mi  r  i  •    i 

hills.  Inat  unseen  power  01  pressure,  which 
resides  in  the  fluid  as  an  organic  law,  exerts  its 
force  with  unceasing  and  untiring  energy.  To 
minds  enlightened  by  science,  and  skill  directed 
by  its  rules,  we  are  indebted  for  one  of  the  no 
blest  works  of  the  present  century.  May  we 
conclusion,  not,  therefore,  conclude  that  science  is  the  only 
sure  means  of  giving  practical  development  to 
those  great  conceptions  which  confer  lasting 
benefits  on  mankind  ?  "  All  that  is  truly  great 
in  the  practical,  is  but  the  result  of  an  antece 
dent  ideal." 


APPENDIX. 


A    COURSE    OF    MATHEMATICS WHAT    IT    SHOULD    BE. 

§  365.  A  COURSE  of  mathematics  should  pre-     A  course 
sent  the  outlines  of  the  science,  so  arranged,  ex-  Mathematics, 
plained,  and  illustrated  as  to  indicate  all  those 
general  methods  of  application,  which  render  it 
effective  and  useful.     This  can  best  be  done  by 
a  series  of  works  embracing  all  the  topics,  and 
in  which  each  topic  is  separately  treated. 

§  366.  Such  a  series  should  be  formed  in  ac-      HOW  it 

cordance  with  a  fixed  plan  ;   should  adopt  and  ^"med.6 
use  the  same  terms  in  all  the  branches ;  should 
be  written  throughout  in   the  same  style ;   and 

present  that  entire  unity  which  belongs  to  the  unity  of  the 
subject  itself.  Bubject> 

§  367.    The  reasonings   of  mathematics   and  Reasonings 
the  processes  of  investigation,   are  the  same  in 


342  APPENDIX. 


the  same  in   every  branch,  and  have  to  be  learned  but  once, 

if  the  same  system  be  studied  throughout.     The 

Different     different  kinds  of  notation,  though  somewhat  un- 

kinds  of  no 
tation,      like  in  the  different  subjects  of  the  science,  are, 

in  fact,  but  dialects  of  a  common  language. 


Language        §  359     if   tneri}   the  language  is,   or  may  be 

need  be 

learned  but  made  essentially  the  same  in  all  the  branches  of 

mathematical  science  ;    and  if  there  is,    as  has 

been  fully  shown,  no  difference  in  the  processes 

in  what     °f  reasoning,  wherein  lies  that  difficulty   in   the 

'difficult"?6  acclu^s^ion  °f  mathematical  knowledge  which  is 
often  experienced  by  students,  and  whence  the 
origin  of  that  opinion  that  the  subject  itself  is 
dry  and  difficult  ? 


A  §  369.  Just  in  proportion  as  a  branch  of  know- 

general  law, 
if  known,    ledge  is  compactly  united  by  a  common  law,  is 

•abject  jN0r.  tne  facmty  of  acquiring  that  knowledge,  if  we 
observe  the  law,  and  the  difficulty  of  acquiring 
Faculties     it,  if  we  pay  no  attention  to  the  law.     The  study 
mauwmatics.  °^   mathematics  demands,  at  every  step,   close 
attention,  nice  discrimination,  and  certain  judg 
ment.      These  faculties  can  only  be  developed 
HOW  first    by  culture.     They  must,  like  other  faculties,  pass 

cultivated:  . 

through  the  states  ot  miancy,  growth,  and  ma 
turity.  They  must  be  first  exercised  on  sensible 
and  simple  objects ;  then  on  elementary  ab- 


APPENDIX.  343 


stract  ideas  ;  and  finally,  on  generalizations  and  on  what 

,,     ,            1-1  finally  exer- 

the  higher  combinations  of  thought  in  the  pure  cised 
ideal. 


§  370.  Have  educators  fully  realized  that  the   Arithmetic 

the  most  im- 

first  lessons  in  numbers  impress  the  first  elements      P0rtant 
of  mathematical    science  ?    that    the  first   con 
nections  of  thought  which  are  there  formed  be 
come  the  first  threads  of  that  intellectual  warp 
which  gives  tone   and  strength   to   the   mind  ? 
Have  they  yet  realized  that  every  process  is,  or     AH  the 
should  be,  like  the  stone  of  an  arch,  formed  to      nected< 
fill,  in  the  entire  structure,   the  exact  place  for 
which  it  is  designed  ?  and  that  the  unity,  beauty, 
and  strength  of  the  whole  depend  on  the  adapta 
tion  of  the  parts   to  each  other  ?      Have   they 
sufficiently  reflected  on  the  confusion  which  must    Necessity 

of  unity  in  all 

arise  from  attempting  to  put  together  and  nar-    the  parts, 
monize  different   parts   of  discordant   systems  ? 
to  blend  portions  that  are  fragmentary,  and  to 
unite  into  a  placid  and  tranquil  stream  trains  of 
thought  which  have  not  a  common  source  ? 

§  371.   Some  have  supposed  that  Arithmetic 
may  be  well  taught  and  learned  without  the  aid 
of  a  text-book ;  or,  if  studied  from  a  book,  that  A  text-book 
the  teacher  may   advantageously  substitute  his 
own  methods  for  those  of  the  author,  inasmuch 


344  APPENDIX. 


tobefoi-  as  such  substitution  is  calculated  to  widen  the 
field  of  investigation,  and  excite  the  mind  of  the 
pupil  to  new  inquiries. 

Reasons.          Admitting  that  every  teacher  of  reasonable 

intelligence,  will  discover  methods  of  communi 

cating  instruction  better  adapted  to  the  peculiar 

ities  of  his  own  mind,  than  all  the  methods  em- 

Evenabet-   ployed  by  the  author  he  may  use;  will  it  be  safe, 

ter  method, 

when  substi-  as  a  general  rule,  to  substitute  extemporaneous 
noth'armo-   methods  for   those   which  have   been  subjected 


to  ^e  analysis  of  science  and  the  tests  of  expe- 
of  the  work.  rjence  ?  Js  it  safe  to  substitute  the  results  of 
known  laws  for  conjectural  judgments  ?  But  if 
they  are  as  good,  or  better  even,  as  isolated  pro 
cesses,  will  they  answer  as  well,  in  their  new 
places  and  connections,  as  the  parts  rejected  ? 
illustration.  Will  the  balance-  wheel  of  a  chronometer  give 
as  steady  a  motion  to  a  common  watch  as  the 
more  simple  and  less  perfect  contrivance  to 
which  all  the  other  parts  are  adapted  ? 

§  372.  If  these  questions  have  significance,  we 
one  of  the   have  found  at  least  one  of  the  causes  that  have 

reasons  why  ..,._.«. 

mathematics  impeded  the  advancement  01  mathematical  sci 
ence,  viz.  the  attempt  to  unite  in  the  same  course 
of  instruction  fragments  of  different  systems  ; 
thus  presenting  to  the  mind  of  the  learner  the 
same  terms  differently  defined,  and  the  same 


APPENDIX.  345 


principles  differently  explained,    illustrated,    and 
applied.      It    is   mutual  relation  and  connection   connection 

very  impor- 

which  bring  sets  of  facts  under  general  laws  ;  it       tant. 
is  mutual  relation  and  connection  of  ideas  which 
form  a  process  of  science ;  it  is  the  mutual  con 
nection    and   relation  of  such   processes  which 
constitute  science  itself. 


§  373.  I  would  by  no  means  be  understood  as    A  teacher 
expressing  the  opinion  that  a  student  or  teacher  should  rei*i 

many  books, 

of  mathematics  should  limit  his  researches  to  a  and  teach  one 

system. 

single  author ;  for,  he  must  necessarily  read  arid 
study  many.  I  speak  of  the  pupil  alone,  who 
must  be  taught  one  method  at  a  time,  and  taught 
that  well,  before  he  is  able  to  compare  different 
methods  with  each  other. 


ORDER  OF  THE  SUBJECTS ARITHMETIC. 

§  374.    Arithmetic   is    the   most   useful   and  Arithmetic: 
simple  branch  of  mathematical  science,  and  is 
the  first  to  be   taught.     If,  however,  the  pupil 
has  time  for  a  full  course,  I  would  by  no  means   connection 
recommend  him  to  finish  his  Arithmetic  before     Aigebra. 
studying  a  portion  of  Algebra. 


346  APPENDIX. 


ALGEBRA. 

Algebra:  §  ^5.  Algebra  is  but  a  universal  Arithmetic, 
with  a  more  comprehensive  notation.  Its  ele 
ments  are  acquired  more  readily  than  the  higher 
and  hidden  properties  of  numbers  ;  and  indeed, 
the  elements  of  any  branch  of  mathematics  are 
more  simple  than  the  higher  principles  of  the 
HOW  preceding  subject  ;  so  that  all  the  subjects  can 

it  should  be  .  . 

studied:     best  be  studied  in  connection  with  those  which 
precede  and  follow. 

should          §  376.  Algebra,  in  a  regular  course  of  instruc- 
tion,  should  precede  Geometry,  because  the  ele 


mentary  processes  do  not  require,  in  so  high  a 

why.       degree,  the  exercise  of  the  faculties  of  abstrac 

tion    and   generalization.      But   when    we   have 

when      completed  the  equation   of  the   second  degree, 

rijouMbe    the  processes  become  more  difficult,  the  abstrac- 

commenced.  tjons  more  perfect,  and  the  generalizations  more 

extended.     Here  then  I  would  pause  and  com 

mence  Geometry. 


GEOMETRY. 


Geometry.  §  377.  Geometry,  as  one  of  the  subjects  of 
mathematical  science/has  been  fully  considered 
in  Book  II.  It  is  referred  to  here  merely  to  mark 
its  place  in  a  regular  course  of  instruction. 


APPENDIX.  347 


TRIGONOMETRY PLANE    AND    SPHERICAL. 

§  378.  The  next  subject  in  order,  after  Geom- 

try : 

etry,  is  Trigonometry :  a  mere  application  of  the 
principles  of  Arithmetic,  Algebra,  and  Geometry   what  it  is. 
to  the  determination  of  the  sides  and  angles  of 
triangles.     As   triangles    are   of  two   kinds,   viz. 
those  formed  by  straight  lines  and  those  formed 
by  the  arcs  of  great  circles  on  the  surface  of  a 
sphere ;    so   Trigonometry  is   divided    into   two  TWO  kinds, 
parts :  Plane  and   Spherical.     Plane   Trigonom 
etry  explains  the  methods,   and  lays  down  the      plane- 
necessary  rules  for  finding  the  remaining  sides 
and  angles  of  a  plane  triangle,  when  a  sufficient 
number  are  known  or  given.     Spherical  Trigo-    spherical, 
nometry  explains  like  processes,  and  lays  down 
similar  rules  for  spherical  triangles. 

SURVEYING  AND  LEVELLING. 

§  379.    The   application   of  the   principles  of 
Trigonometry  to   the   measurement  of  portions 
of  the  earth's  surface,  is  called  Surveying;  and    surveying. 
similar  applications  of  the  same  principles  to  the 
determination  of  the  difference  between  the  dis 
tances  of  any  two  points  from  the  centre  of  the 
earth,  is  called  Levelling.    These  subjects,  which    Levelling, 
follow  Trigonometry,  not  only  embrace  the  va- 


348  APPENDIX. 


what  they  rious  methods  of  calculation,  but  also  a  descrip 
tion  of  the  necessary  Instruments  and  Tables. 
They  should  be  studied  immediately  after  Trigo 
nometry  ;  of  which,  indeed,  they  are  but  appli 
cations. 


DESCRIPTIVE     GEOMETRY. 

Descriptive        §  380.    Descriptive  Geometry  is   that  branch 

Geometry : 

of  mathematics  which  considers  the  positions  of 
the  geometrical  magnitudes,  as  they  may  exist  in 
space,  and  determines  these  positions  by  refer 
ring  the  magnitudes  to  two  planes  called  the 
Planes  of  Projection. 

its  nature.  It  is,  indeed,  but  a  development  of  those  gen 
eral  methods,  by  which  lines,  surfaces,  and  solids 
may  be  presented  to  the  mind  by  means  of 
drawings  made  upon  paper.  The  processes  of 
what  its  this  development  require  the  constant  exercise  of 

StUpTisheCs°ra~  tne  conceptive  faculty.  All  geometrical  mag 
nitudes  may  be  referred  to  two  planes  of  pro 
jection,  and  their  representations  on  these  planes 
will  express  to  the  mind,  their  forms,  extent,  and 
also  their  positions  or  places  in  space.  From 
HOW.  these  representations,  the  mind  perceives,  as  it 
were,  at  a  single  view,  the  magnitudes  them 
selves,  as  they  exist  in  space  ;  traces  their  boun 
daries,  measures  their  extent,  and  sees  all  their 
parts  separately  and  in  their  connection. 


APPENDIX.  349 


In  France,  Descriptive  Geometry  is  an  impor-       HOW 

,  ,  regarded  in 

tant  element  of  education.     It  is  taught  in  most      Frauce> 
of  the  public  schools,  and  is  regarded  as  indis 
pensable  to  the  architect  and  engineer.     It  is, 
indeed,  the  only  means  of  so  reducing  to  paper, 
and  presenting  at  a  single  view,  all  the  compli 
cated  parts  of  a  structure,  that  the  drawing  or 
representation  of  it  can  be  read  at  a  glance,  and 
all  the  parts  be  at  once  referred  to  their  appropri 
ate  places.     It  is  to  the  engineer  or  architect  not     its  value 
only  a  general  language  by  which  he  can  record     branch. 
and  express   to  others  all  his  conceptions,  but  is 
also  the  most  powerful  means  of  extending  those 
conceptions,  and  subjecting  them  to  the  laws  of 
exact  science. 


SHADES,    SHADOWS,    AND    PERSPECTIVE. 

§  381.  The  application  of  Descriptive  Geom 
etry  to  the  determination  of  shades  and  shadows,      shades, 

Shadows, 

as  they  are  found   to   exist  on  the  surfaces  of       and 
bodies,  is  one  of  the  most  striking  and  useful  ap-  persPectlve- 
plications  of   science  ;    and  when  it   is  further 
extended  to  the  subject  of  Perspective,  we  have 
all  that  is  necessary  to  the  exact  representation 
of  objects  as  they  appear  in  nature.     An  accu 
rate  perspective   and   the   right   distribution   of 
light  and  shade  are  the  basis  of  every  work  of 


3"0  APPENDIX 


mm  the  fine  arts.  Without  them,  the  sculptor  ar.d 
the  painter  would  labor  in  vain :  the  chisel  of 
Canova  would  give  no  life  to  the  marble,  nor  the 
touches  of  Raphael  to  the  canvas. 


ANALYTICAL    GEOMET1Y. 

f  382.  Analytical  Geometry  is  the  next  sub 
ject  in  a  regular  course  of  mathematical  study. 
though  it  may  be  studied  before  Descriptive  Ge 
ometry.  The  importance  of  this  subject  cannut 
be  exaggerated.  In  Algebra,  the  symbols  of 
quantity  have  generally  so  close  a  connection 
with  numbers,  that  the  mind  scarcely  realizes 
»  the  extent  of  the  generalization  :  and  the  power 
of  analysis,  arising  from  the  changes  that  mav 
take  place  among  the  quantities  which  the  sym 
bols  represent,  cannot  be  fully  explained  and  de- 


But  in  Analytical  Geometry,  where  all  the 
magnitudes  axe  brought  under  the  power  of  anal- 
jmm,  and  all  their  properties  developed  by  the 
combined  processes  of  Algebra  and  Geometry,  we 
are  brought  to  feel  the  extent  and  potency  of 
those  methods  which  combine  in  a  single  equa 
tion  every  discovered  and  undiscovered  property 
of  every  fine,  straight  or  curved,  which  can  be 
tin  ii  Mil  by  the  intersection  of  a  cone  and  plane. 


APPENDIX.  3". 

To  develop  every  property  of  the  Conic  Sec-  itsextoi. 
tions  from  a  single  equation,  and  that  an  equa 
tion  only  of  the  second  degree,  by  the  known 
processes  of  Algebra,  and  thus  interpret  the  re 
sults,  is  a  far  different  exercise  of  the  mind  from 
that  which  arises  from  searching  them  out  by 
the  tedious  and  disconnected  methods  of  separate 
propositions.  The  first  traces  all  from  an  inex-  i»  methods 
haustible  fountain  by  the  known  laws  of  analyti 
cal  investigation,  applicable  to  all-  similar  cases, 
while  the  latter  adopts  particular  processes  ap 
plicable  to  special  cases  only,  without  any  gen 
eral  law  of  connection. 


DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

§383.  The  Differential  and  Integral  Calculus 
presents  a  new  view  of  the  power,  extent,  and 
applications  of  mathematical  science.  It  should 
be  carefully  studied  by  all  who  seek  to  make 
high  attainments  in  mathematical  knowledge,  or 
who  desire  to  read  the  best  works  on  Natural 
and  Experimental  Philosophy.  It  is  that  field  of 
mathematical  investigation,  where  genius  may 
exert  its  highest  powers  and  find  its  most  certain 
rewards. 


INDEX. 


ABSTRACTION That  faculty  of  the  mind  which  enables  us,  in  contem 
plating  any  object  to  attend  exclusively  to  some  par 
ticular  circumstance,  and  quite  withhold  our  attention 
from  the  rest,  Section  12. 
"  Is  used  in  three  senses,  13. 

Abstract  Quantity,  75,  96. 

Addition, Readings  in,  116. 

Examples  in,  151. 

"  of  Fractions,  Rule  for,  191. 

"  Combinations  in,  192, 193. 

Definitions  of,  203. 
"  One  principle  governs  all  operations  in,  232. 

^Etna, How  far  designated  by  the  term  mountain,  20. 

A  Geometrical       Proportion,  168. 

ALGEBRA A  species  of  Universal  Arithmetic,  in  which  letters  and 

signs  are  employed  to  abridge  and  generalize  all  pro 
cesses  involving  numbers,  280. 
"  Divided  into  two  parts,  280. 

"  Difficulties  of,  from  what  arising,  286. 

"  Principles  of,  deduced  from  definitions  and  axioms,  297. 

"  Should  precede  Geometry  in  instruction,  376. 

Alphabet  of  the  language  of  numbers,  80,  113,  114. 

"  Language  of  Arithmetic,  formed  from,  192. 

Analytical  Form,  for  what  best  suited,  71,  89. 

ANALYSIS A  term  embracing  all  the  operations  that  can  be  performed 

on  quantities  represented  by  letters,  87,  88,  274,  327. 
"  It  also  denotes  the  process  of  separating  a  complex  whole 

into  its  parts,  89. 

44  of  problems  in  Arithmetic,  175,  176. 

23 


354 


ANALYSIS. 


INDEX. 


Angles.. 


Apothecaries' 
APPREHENSION. 


AREA  or .. 
Argument 


Arguments, 


Aristotle 


ARISTOTLE'S 


.Three  branches  of,  Sections  279,  285,  286. 
First  notions  of,  how  acquired,  317. 
Problems  it,  has  solved,  344-347. 
.Right  angle,  the  unit  of,  250. 
A  class  of  Geometrical  Magnitudes,  273. 
Weight — Its  units  and  scale,  138. 
..Simple  apprehension  is  the  notion  (or  conception)  of  an 

object  in  the  mind,  7. 

Incomplex  apprehension  is  of  one  object  or  of  several 
without  any  relation  being  perceived  between  them,  7. 
Complex  is  of  several  with  such  a  relation,  7. 
.CONTENTS,  Number  of  times  a  surface  contains  its  unit  of 

measure,  141. 

with  one  premiss  suppressed  is  called  an  Enthymeme,  47. 
Two  kinds  of  objections  to  an,  47. 
Every  valid,  may  be  reduced  to  a  syllogism,  52. 
at  full  length,  a  syllogism,  56. 

concerned  with  connection  between  premises  and  conclu 
sion,  57. 

Where  the  fault  (if  any)  lies,  69. 
In  reasoning  we  make  use  of,  42. 
Examples  of  unsound,  50. 
Rules  for  examining,  70. 
did  not  mean  that  arguments  should  always  be  stated 

syllogistically,  53. 
accused  of  darkening  his  demonstrations  by  the  use  of 

symbols,  57 

His  philosophy  not  progressive,  334. 
DICTUM — Whatever  is  predicated  (that  is,  affirmed  or  de 
nied)  universally,  of  any  class  of  things,  may 
be  predicated,  in  like  manner  (viz.  affirmed 
or  denied),  of  any  thing  comprehended  in 
that  class,  54. 

"          Keystone  of  his  logical  system,  54. 
"          Objections  to,  54,  55. 

a  generalized  statement  of  all  demonstration,  55. 
"          applied  to  terms  represented  by  letters,  56. 
"          not  complied  with,  59,  60. 
"          All  sound  arguments  can  be  reduced  to  the  form 
to  which  it  applies,  65,  66. 


INDEX. 


355 


ARITHMETIC. 


Arithmetical 
ART 


.Is  both  a  science  and  an  art,  Section  172. 
It  is  a  science  in  all  that  relates  to  the  properties,  laws, 

and  proportions  of  numbers,  172. 
It  is  an  art  in  all  that  concerns  their  application,  173. 
Processes  of,  not  affected  by  the  nature  of  the  objects,  43. 
Illustration  from,  45. 

How  its  principles  should  be  explained,  174 
Its  requisitions  as  an  art,  177. 
Faculties  cultivated  by  it,  180. 
Application  of  principles,  188. 
Generally  preceded  by  a  smaller  treatise,  190, 
Methods  of  placing  subjects  examined,  191. 
Combinations  in,  192-199. 
What  its  study  should  accomplish,  206. 
Art  of,  its  importance,  206. 

Elementary  ideas  of,  learned  by  sensible  objects,  207. 
Principles  of,  how  they  should  be  taught,  208. 
FIRST,  what  it  should  accomplish,  214. 

"     arrangement  of  lessons,  214-223. 

"      what  should  be  taught  hi  it,  226. 
SECOND,  should  be  complete  and  practical,  227. 

"         arrangement  of  subjects,  228. 

"        introduction  of  subjects,  229. 

"        reading   of  figures   should   be   constantly  prac 
tised,  230. 
THIRD,  the  subject  now  taught  as  a  science,  231. 

"       requirements  from  the  pupil  for,  231. 

"       Reduction  and  the  ground  rules  brought  under  one 
principle,  232. 

"       design  of,  —  methods   must  differ   from    smaller 
works,  233. 

"       examples  in  the  ground  rules,  234. 

"      what  subjects  should  be  transferred  from  elemen 
tary  works,  235. 
Practical  utility  of,  356,  357. 

should  not  be  finished  before  Algebra  is  commenced,  374. 
Proportion,  163. 
Ratio,  163. 

.The  application  of  knowledge  to  practice,  22. 
Its  relations  to  science,  22. 


356 


INDEX. 


ART. A  single  one  often  formed  from  several  sciences,  Section  2  2. 

of  Arithmetic,  173,  177,  182. 
Astronomy  brought  by  Newton  within  the  laws  of  mechanics,  337. 

How  it  became  deductive,  339. 

Mathematics  necessary  in,  341. 
Authors,  methods  of  finding  ratio,  165,  170. 

"         of  placing  Rule  of  Three,  187. 

quotations  from,  on  Arithmetic,  201-204 

definition  of  proportion,  268. 
Auxiliary  Quantities,  259,  261. 

Avoirdupois  Weight,  its  units  and  scale,  136 

AXIOM A  self-evident  truth,  27,  97. 

Axioms  of  Geometry,  process  of  learning  them,  27. 

or  canons,  for  testing  the  validity  of  syllogisms,  67. 

of  Geometry  established  by  Induction,  73. 

for  forming  numbers,  79. 

for  comparison  relate  to  equality  and  inequality,  102. 
"  for  inferring  equality,  102,  258,  260,  264. 

"  "         "         inequality,  102. 

employed  in  solving  equations,  278,  311. 

Bacon,  Lord,  Quotation  from,  328. 

Foundation  of  his  Philosophy,  334  ;  its  subject  Nature, 

335,  page  12. 

"  His  system  inductive,  334. 

"  Object  and  means  of  his  philosophy,  335 

Barometer,  Construction  and  use  of,  359 

Barrow,  Dr.,  Quotation  from,  328,  340. 

Belief  essential  to  knowledge,  23. 

"  and  disbelief  are  expressed  in  propositions,  36. 

Blakewell,  steps  of  his  discovery,  32. 

Bowditch,  Tables  of,  used  in  Navigation,  359. 

BREADTH A  dimension  of  space,  82. 

Bridge,  Harlem,  description  of,  362. 

CALCULUS, In  its  general  sense,  means  any  operation  performed  on 

algebraic  quantities,  281,  282. 

"  Differential  and  Integral,  283-285,  383. 

Canons  for  testing  the  validity  of  syllogisms,  67. 

Cause  and  effect,  their  relation  the  scientific  basis  of  induction,  33. 


INDEX. 


357 


Chemist,  Illustration,  Section  53  ;  idea  of  iron,  322. 

Chemistry  aided  by  Mathematics,  342. 

CIRCLE A  portion  of  a  plane  included  within  a  curve,  all  the 

points  of  which  are  equally  distant  from  a  certain  point 
within  called  the  centre,  244. 

"  The  only  curve  of  Elementary  Geometry,  244. 

"  Property  of,  256. 

Circular  Measure,  its  units  and  scale,  149. 

CLASSES Divisions  of  species  or  subspecies,  in  which  the  charac 
teristic  is  less  extensive,  but  more  full  and  complete,  16. 
CLASSIFICATION..... The  arrangement  of  objects  into  classes,  with  reference  to 

some  common  and  distinguishing  characteristic,  16. 
"  Basis  of,  may  be  chosen  arbitrarily,  20. 

Coefficient  of  a  letter,  291 ;  of  a  product,  292. 

Differential,  283,  284. 

Coins  should  be  exhibited  to  give  ideas  of  numbers,  133. 

Combinations          in  Arithmetic,  192-199. 

"  taught  in  First  Arithmetic,  216-218. 

Comets,  Problem  with  reference  to,  347. 

Comparison,  Knowledge  gained  by,  95. 

"  Reasoning  carried  on  by,  25,  307. 

CONCLUSION The  third  proposition  of  a  syllogism,  40. 

"  in  Induction,  broader  than  the  premises,  31. 

"  deduced  from  the  premises,  40,  41,  46,  47,  49. 

"  contradicts  a  known  truth,  in  negative  demonstrations, 

264,  265. 

Concrete  Quantity,  75,  96. 

Conjunctions  causal,  illative,  48. 

"  denote  cause  and  effect,  premiss  and  conclusion,  48. 

CONSTANTS    Quantities  which  preserve  a  fixed  value  throughout  the 

same  discussion  or  investigation,  282,  283,  313. 
"  represented  by  the  first  letters  of  the  alphabet,  284 

COPULA  That  part  of  a  proposition  which  indicates  the  act  of 

judgment,  38. 

"  must  be  "is"  or  "is  not,"  38,  39. 

Cousin,  quotation  from,  180. 

Curves,  circumference  of  circle  the  simplest  of,  239. 

Croton  river,  its  sources,  362. 

"  dam,  its  construction,  362  ;  lake,  area  of,  362 

aqueduct,  description  of,  362. 


358 


INDEX. 


Decimals,                language  and  scale  for,  Sections  156,  157. 
DEDUCTION  A  process  of  reasoning  by  which  a  particular  truth  is  in 
ferred  from  other  truths  which  are  known  or  admitted,  34. 
"  Its  formula  the  syllogism,  34. 

Deductive  Sciences,  why  they  exist,  98. 

"  "          aid  they  give  in  Induction,  335. 

DEFINITION  A  metaphorical   word,  which  literally   signifies    laying 

down  a  boundary,  1. 
"  Is  of  two  kinds,  1. 

"  Its  various  attributes,  2-5. 

Definitions,  General  method  of  framing,  3. 

"  Rules  for  framing,  5  (Note). 

"  and  axioms,  tests  of  truth,  97,  99. 

"  signs  of  elementary  ideas,  200. 

"  Necessity  of  exact,  200. 

DEMONSTEATION  ...A  series  of  logical  arguments  brought  to  a  conclusion,  in 
which  the  major  premises  are  definitions,  axioms,  or 
propositions  already  established,  237. 
"  of  a  demonstration,  55. 

"  to  what  applicable,  238. 

"  of  Proposition  I.  of  Legendre,  258. 

"  positive  and  negative,  262-265. 

"  produces  the  most  certain  knowledge,  326. 

Descartes,  originator  of  Analytical  Geometry,  281 

Dictum,  Aristotle's,  54,  55,  66. 

DIFFERENTIAL  AND  INTEGRAL  CALCULUS.  The  science  which  notes  the 
changes  that  take  place  according  to  fixed  laws  estab 
lished  by  algebraic  formulas,  when  those  changes  ;ir<: 
indicated  by  certain  marks  drawn  from  the  variable 
symbols,  283. 
Coefficients — Marks  drawn  from  the  variable  symbols, 

283,  284. 

and  Integral  Calculus — Difference  between  it  and  Ana 
lytical  Geometry,  284. 

"  "  "  "  "What  persons  should  study  it,  383. 

Discussion  of  an  Equation,  308. 

DISTRIBUTION  A  term  is  distributed,  when  it  stands  for  all  its  signifi- 

cates,  61. 

g 

A  term  is  not  distributed  when  it  stands  for  only  a  part 
of  its  significates,  61. 


INDEX.  359 


Distribution,  Words  which  mark,  not  always  expressed,  Section  62. 

Division,  Readings  in,  123  ;  examples  in,  154. 

"  Combinations  in,  196, 

"  All  operations  in,  governed  by  one  principle,  232. 

"  of  quantities,  how  indicated,  294. 

Dry  Measure,  Its  units  and  scale,  147. 

Duodecimal  units,  142-144. 

English  Money,       Its  units  and  scale,  135. 

ENTHYMEME An  argument  with  one  premiss  suppressed,  47. 

EQUAL. Two  geometrical  figures  are  said  to  be  equal  when  they 

can  be  so  applied  to  each  other  as  to  coincide  through 
out  their  whole  extent,  255,  312. 

EQUALITY In  Geometry  expresses   that  two   figures  coincide.     In 

Algebra  it  merely  implies  that  each  member  of  an 
equation  contains  the  same  unit  an  equal  number  of 
times,  312. 

EQUATION An  analytical  formula  for  expressing  equality,  307-312. 

"  A  proposition  expressed  algebraically,  in  which  equality 

is  predicated  of  one  quantity  as  compared  with  an 
other,  309. 
"  either  abstract  or  concrete,  310. 

Equations,  subject  of,  divided  into  two  parts,  308. 

Five  axioms  for  solving,  311. 

EQUIVALENT Two  geometrical  figures  are  said  to  be  equivalent  when 

they  contain  the  same  unit  of  measure  an  equal  num 
ber  of  times,  255. 

Examples  in  ground  rules  of  Third  Arithmetic,  234. 

Of  little  use  to  vary  forms  of,  without  changing  the  prin 
ciples  of  construction,  236. 

Experiment,  in  what  sense  used,  25  (Note). 

EXPONENT An  expression  to  show  how  many  equal  factors  are  em 
ployed,  293. 

Extremes.  Subject  and  predicate  of  a  proposition,  38,  67. 

FACT Any  thing  which  has  been  or  is,  24. 

"  Knowledge  of,  how  derived,  25. 

"  In  what  sense  used,  25. 

"  regarded  as  a  genus,  25. 

Factories,  value  of  science  in,  358. 


360 


INDEX 


FALLACY Any  unsound  mode  of  arguing  which  appears  to  demand 

our  conviction,  and  to  be  decisive  of  the  question  in 
hand,  when  in  fairness  it  is  not,  Section  68. 
Illustration  of,  53. 

"  Example  and  analysis  of,  59,  60. 

"  Material  and  Logical,  69. 

"  Rules  for  detecting,  70. 

Federal  Money,       units  increase  by  scale  of  tens,  129,  134. 

"  Methods  of  reading,  129,  134. 

FIGURE A  portion  of  space  limited  by  boundaries,  83. 

"  Each  geometrical,  stands  for  a  class,  277. 

Figures  in  Arithmetic  show  how  many  times  a  unit  is  taken,  125. 

do  not  indicate  the  kind  of  unit,  125. 
Laws  of  the  places  of,  126, 127. 

"  have  no  value,  128,  201. 

"  Methods  of  reading,  130  ;  of  writing,  199. 

"  Definitions  of,  201,  202. 

"  should  be  early  used  in  Arithmetic,  219. 

First  Arithmetic,    what  should  be  taught  in  it,  226. 

"  Faculties  to  be  cultivated  by  it,  214. 

"  Construction  of  the  lessons,  214-218. 

"  Lesson  in  Fractions,  220-224. 

"  Tables  of  Denominate  Numbers — Examples,  225. 

Fractions  come  from  the  unit  one,  132. 

"  should  be  constantly  compared  with  one,  162. 

"  Reasons  for  placing  Common  Fractions  immediately  after 

Division  examined,  189. 

"  not  "  unexecuted  divisions,"  189. 

"  Elementary  idea  of,  189. 

"  Expression  for,  the  same  as  for  Division,  189. 

«  Definitions  of,  204. 

"  Lessons  in,  in  First  Arithmetic,  220-224. 

FRACTIONAL  units,  155  ;  orders  of,  156  ;  language  of,  156-159, 197. 

"  "     three  things  necessary  to  their  apprehension,  160. 

«  "     advantages  of,  161. 

"  "     two  things  necessary  to  their  being  equal,  161. 

Galileo,  imprisoned  in  the  17th  century,  343. 

GENERALIZATION.... The  process  of  contemplating  the  agreement  of  several 
objects  in  certain  points,  and  giving  to  all  and  each  of 


INDEX.  36] 


these  objects  a  name  applicable  to  them  in  respect  to 
this  agreement,  Section  14. 
Generalization        implies  abstraction,  14. 

"  must  be  preceded  by  knowledge,  1 84. 

GENUS The   most  extensive   term  of  classification,   and   conse 
quently  the  one  involving  the  fewest  particulars,  16,  17. 
"  HIGHEST.     That  which  cannot  be  referred  to  a  more  ex 

tended  classification,  19. 

"  SUBALTERN.     A  species  of  a  more  extended  classifica 

tion,  18. 

Geometrical  Magnitudes,  three  classes  of,  238,  273. 

"  "  do  not  involve  matter,  247. 

"  "  their  boundaries  or  limits,  247. 

"  "  each  has  its  unit  of  measure,  252. 

"  "  analysis  of  comparison,  270,  271. 

"  "  to  what  the  examination  of  properties  has 

reference,  273. 
"  Proportion,  163  ;  Ratio,  163  ;  Progression,  170. 

GEOMETRY Treats  of  space,  and  compares  portions  of  space  with  each 

other,  for  the  purpose  of  pointing  out  their  properties 
and  mutual  relations,  237. 
*  Why  a  deductive  science,  257. 

«  First  notions  of,  how  acquired,  318-320. 

«  Practical  utility  of,  357. 

u  Origin  of  the  science,  360. 

"  Its  place  in  a  course  of  instruction,  377. 

tt  ANALYTICAL,  Examines  the  properties,  measures,  and  re 

lations  of  the  Geometrical  Magnitudes  by 
means  of  the  analytical  symbols,  281,  282. 
«  "  originated  with  Descartes,  281. 

u  "  difference  between  it  and  Calculus,  284. 

"  "  its  importance,  extent,  and  methods,  382. 

"  DESCRIPTIVE.     That  branch  of  mathematics  which  con 

siders  the  positions  of  the  Geometrical 
Magnitudes  as  they  may  exist  in  space, 
and  determines  these  positions  by  re 
ferring  the  magnitudes  to  two  planes 
called  the  Planes  of  Projection,  380. 

«  «  how  regarded  in  France,  380. 

Governor,  functions  of,  in  machinery,  358. 


332 


INDEX. 


Grammar 
Gravitation, 


defined,  Section  113. 
Law  of,  32,  344. 


Hall,  Captain's,      voyage  from  San  Bias  to  Rio  Janeiro,  359. 
Harlem  river,          Bridge  over,  and  width,  362. 
Herschel,  Sir  John,  Quotation  from,  27,  322,  341,  359. 
Hull  of  the  steamship,  how  formed,  359. 

Illative  Conjunctions,  48. 

ILLICIT  PROCESS When  a  term  is  distributed  in  the  conclusion  which  was 

not  distributed  in  one  of  the  premises,  67. 
Indefinite  Propositions,  62. 

Index  of  a  root,  295. 

INDUCTION  Is  that  part  of  Logic  which  infers  truths  from  facts,  30-33. 

Logic  of,  30. 

"  supposes  necessary  observations  accurately  made,  32. 

"  Example  of,  Blakewell,  32  ;  of  Newton,  32. 

"  based  upon  the  relation  of  cause  arid  effect,  33. 

u  Reasoning  from  particulars  to  generals,  34. 

"  its  place  in  Logic,  72. 

"  how  thrown  into  the  form  of  a  syllogism,  74,  99. 

"  Truths  of,  verified  by  Deduction,  335,  336. 

Inertia  proportioned  to  weight,  268. 

INFINITY, The  limit  of  an  increasing  quantity,  302-306. 

Integer  Numbers,  why  easier  than  fractions,  162. 

"  constructed  on  a  single  principle,  231. 

INTUITION Is  strictly  applicable  only  to  that  mode  of  contemplation, 

in  which  we  look  at  facts,  or  classes  of  facts,  and  im 
mediately  apprehend  their  relations,  27. 
Iron,  different  ideas  attached  to  the  word,  322. 

JUDGMENT  Is  the  comparing  together  in  the  mind  two  of  the  notions 

(or  ideas)  which  are  the  objects  of  apprehension,  and 
pronouncing  that  they  agree  or  disagree,  8. 
"  is  either  Affirmative  or  Negative,  8. 

Kant,  quotation  from,  21. 

KNOWLEDGE  Is  a  clear  and  certain  conception  of  that  which  is  true,  23. 

"  facts  and  truths  elements  of,  25. 

of  facts,  how  derived,  25. 


INDEX. 


*363 


Knowledge 


some  possessed  antecedently  to  reasoning,  Section  29. 

the  greater  part  matter  of  inference,  29. 

must  precede  generalization,  184. 

two  ways  of  increasing,  323. 

cannot  exceed  our  ideas,  323. 

the  increase  of,  renders  classification  necessary,  page  20. 


LANGUAGE 


Affords  the  signs  by  which  the  operations  of  the  mind  are 

recorded,  expressed,  and  communicated,  10. 
"  Every  branch  of  knowledge  has  its  own,  11. 

"  of  numbers,  80;  of  mathematics,  91,  92. 

•'  of  mathematics  must  be  thoroughly  learned,  92. 

«  "          "  its  generality,  93. 

"  for  fractional  units,  156,  159,  197. 

«  Arithmetical,  192-199. 

"  exact,  necessary  to  accurate  thought,  205. 

"  of  Arithmetic,  its  uses,  219. 

«  of  Algebra,  the  first  thing  to  which  the  pupil's   mind 

should  be  directed,  290. 

"  Culture  of  the  mind  by  the  use  of  exact,  322. 

Laws  of  Nature,     Science  makes  them  known,  21,  315. 

«  "       refers  individual  cases  to  them,  55. 

«  generalized  facts,  55,  page  14. 

"  include  all  contingencies,  332. 

"  every  diversity  the  effect  of,  346. 

one  dimension  of  space,  8 1. 
in  First  Arithmetic,  how  arranged,  214. 
"     "  "  their  connections,  218. 

may  stand  for  all  numbers,  276. 
"  represents  things  in  general,  277. 

LEVELLING  The  application  of  the  principles  of  Trigonometry  to  the 

determination  of  the  difference  between  the  distances 
of  any  two  points  from  the  centre  of  the  earth,  379. 
"  Its  practical  uses,  360. 

Limit,  definition  of,  306. 

LINE One  dimension  of  space,  83,  239. 

"  A  straight  line  does  not  change  its  direction,  83,  239,  318. 

"  Curved  line,  one  which  changes  its  direction  at  every 

point,  83,  239. 
«  Axiom  of  the  straight,  239. 


Length 
Lessons 

Letter 


364  INDEX. 


Lines,  limits  of,  Section  247 

"  Auxiliary,  259. 

Liquid  Measure,     Its  units  and  scale,  146. 
"  Local  value  of  a  figure,"  has  no  significance,  128,  201. 
Locke,  Quotation  from,  323. 

LOGIC  .: Takes  note  of  and  decides  upon  the  sufficiency  of  the  evi 
dence  by  which  truths  are  established,  29. 

"  Nearly  the  whole  of  science  and  conduct  amenable  to,  29. 

"  of  Induction,  its  nature,  30. 

"  Archbishop  Whateley's  views  of,  72. 

"  Mr.  Mill's  views  of,  72. 

Logical  Fallacy,      69. 

Machinery  of  factories  arranged  on  a  general  plan,  358. 

of  the  steamship,  359. 
Major  Premiss,       often  suppressed,  cannot  be  denied,  46. 

"  ultimate,  of  Induction,  74,  99. 

Major  Premises      of  Geometry,  237,  257. 
Mansfield,  Mr.,        Quotation  from,  325,  327. 

MARK The  evidence  contained  in  the  attributes  implied  in  a 

general  name,  by  which  we  infer  that  any  thing  called 
by  that  name  possesses  another  attribute  or  set  of  at 
tributes.  For  example  :  "  All  equilateral  triangles  are 
equiangular."  Knowing  this  general  proposition,  when 
we  consider  any  object  possessing  the  attributes  implied 
in  the  term  "  equilateral  triangle,"  we  may  infer  that  it 
possesses  the  attributes  implied  in  the  term  "  equian 
gular;"  thus  using  the  first  attributes  as  a  mark  or 
evidence  of  the  second.  Hence,  whatever  possesses 
any  mark  possesses  those  attributes  of  which  it  is  a 
mark,  98,  257  259. 

Masts  of  the  steamship,  how  placed,  359. 

Material  Fallacy,    69. 
Mathematical          Reasoning  conforms  to  logical  rules,  73. 

every  truth  established  by,  is  developed  by  a 
process  of  Arithmetic,  Geometry,  or  Analy 
sis,  or  a  combination  of  them,  90. 

MATHEMATICS  The  science  of  quantity,  76. 

"  Pure,  embraces  the  principles  of  the  science,  76-78. 

"      on  what  based,  97. 


INDEX. 


365 


MATHEMATICS  Mixed,  embraces  the  applications,  Section  76. 

"  Primary  signification,  77. 

"  Language  of,  91. 

*  "  Exact  science,"  97. 

"  Logical  test  of  truth  in,  97. 

"  a  deductive  science,  97,  98. 

u  concerned  with  number  and  space,  73,  76,  78,  101. 

"  What  gives  rise  to  its  existence,  100. 

"  Why  peculiarly  adapted  to  give  clear  ideas,  324-326,  329 

"  a  pure  science,  329. 

"  considered  as  furnishing  the  keys  of  knowledge,  331. 

"  Widest  applications  are  in  nature,  334. 

«  Effects  on  the  mind  and  character,  328,  340. 

«  Guidance  through  Nature,  340. 

"  Its  necessity  in  Astronomy,  341. 

«  Results  reached  by  it,  349,  350. 

*  Practical  advantages  of,  355. 

"  What  a  course  of,  should  present,  and  how,  365,  366. 

«  Reasonings  of,  the  same  in  each  branch,  367. 

"  Faculties  required  by,  369. 

«  Necessity  of,  to  the  philosopher,  page  16. 

MEASURE A  term  of  comparison,  94. 

«  Unit  of.  should  be  exhibited  to  give  ideas  of  numbers,  133. 
«  "      for  lines,  surfaces,  solids,  249. 

u  of  a  magnitude,  how  ascertained,  249. 

Middle  Term  distributed  when  the  predicate  of  a  negative   proposi 
tion,  64. 

"  When  equivocal,  67. 

Mill,  Mr.  his  views  of  Logic,  72,  74. 

Mind,  Operations  of,  in  reasoning,  6. 

«  Abstraction  a  faculty,  process,  and  state  of,  13. 

«  Processes  of,  which  leave  no  trace,  68. 

«  Faculties  of,  cultivated  by  Arithmetic,  180. 

«  Thinking  faculty  of,  peculiarly  cultivated  by  mathemat 
ics,  325,  326. 

Minus  sign,  Power  of,  fixed  by  definition,  297. 

Motion  proportional  to  force  impressed,  268. 
Multiplication,        Readings  in,  122  ;  examples  in,  153. 

«  What  the  definition  of,  requires,  177. 

«  Combinations  in,  195. 


366  INDEX. 


Multiplication,        All  operations  in,  governed  by  one  principle,  Section  232. 
"  in  Algebra,  illustrations  of,  299-301. 

Names,  Definitions  are  of,  1. 

"  given  to  portions  of  space,  and  defined  in  Geometry,  238. 

Naturalist  determines  the  species  of  an  animal  from  examining  a 

bone,  333. 
Negative  premises,  nothing  can  be  inferred  from,  67. 

"  demonstration,  its  nature,  263,  265  ;  illustration  of,  264. 

Newton,  his  method  of  discovery,  32. 

"  changed  Astronomy  from  an  experimental  to  a  deductive 

science,  337,  339. 
Non-distribution     of  terms,  61. 

"  Word  "  some"  which  marks,  not  always  expressed,  62 

NUMBERS Are  expressions  for  one  or  more  things  of  the  same  kind, 

79,  106. 

"  How  learned,  79. 

"  Axioms  for  forming,  79,  304. 

"  Three  ways  of  expressing,  107. 

u  Ideas  of,  complex,  108,  124. 

"  Two  things  necessary  for  apprehending  clearly,  110. 

"  Simple  and  Denominate,  112. 

0  Examples  of  reading  Simple,  130. 

"  Two  ways  of  forming  from  ONE,  131. 

"  first  learned  through  the  senses,  133,  316. 

"  Two  ways  of  comparing,  163. 

"  compared,  must  be  of  the  same  kind,  171,  175. 

«  Definitions  of,  201,  202. 

"  must  be  of  something,  275. 

«  may  stand  for  all  things,  276. 

"  '       First  lessons  in,  impress  the  first  elements  of  mathemati 

cal  science,  370. 

Olmsted's  Mechanics,  quotation  from,  269. 

Optician,  Illustration,  212. 

Oral  Arithmetic,  its  inefficiency  without  figures,  219. 

Order  of  subjects  in  Arithmetic,  182,  188. 

PARALLELOGRAM  ...A  quadrilateral  having  its  opposite  sides  taken  two  and 
two  parallel,  242. 


INDEX.  367 


Parallelogram         regarded  as  a  species,  Section  17  ;  as  a  genus,  18. 

"  Properties  of,  256. 

Particular  proposition,  62. 

"  premises,  nothing  can  be  proved  from,  67. 

Pendulum,  the  standard  for  measurement,  253. 

Philosophy,  Natural,  originally  experimental,  337. 

"        has  been  rendered  mathematical,  337. 
Place  idea  attached  to  the  word,  81. 

"  designates  the  unit  of  a  number,  202. 

PLANE That  with  which  a  straight  line,  having  two  points  in 

common,  and  any  how  placed,  will  coincide,  240. 
"  First  idea  of,  how  impressed,  319. 

PLANE  FIGURE  ...  .Any  portion  of  a  plane  bounded  by  linen,  240. 
Plane  Figures         in  general,  243. 

POINT That  which  has  position  in  space  without  occupying  any 

part  of  it,  81. 

Points,  extremities  or  limits  of  a  line,  239. 

Practical  Rules  in  Arithmetic,  177,  178. 

"  The  true,  207,  must  be  the  consequent  of  science,  228. 

"  Popular  meaning  of,  351,  353. 

"  Questions  with  regard  to,  351,  352. 

"  Consequences  of  an  erroneous  view  of,  354. 

"  True  signification  of,  354. 

Practice  precedes  theory,  but  is  improved  by  it,  42, 

"  without  science  is  empiricism,  page  13. 

PREDICATE That  which  is  affirmed  or  denied  of  the  subject,  38 

"  Distribution,  63. 

«  Non -distribution,  63. 

"  sometimes  coincides  with  the  subject,  63. 

PREMISS Each  of  two  propositions  of  a  syllogism  admitted  to  be 

true,  40. 
MAJOR  PREMISS — The  proposition  of  a  syllogism  which 

contains  the  predicate  of  the  conclusion,  40. 
MINOR  PREMISS — The  proposition  of  a  syllogism  which 

contains  the  subject  of  the  conclusion,  40. 
Pressure,  a  law  of  fluids,  364. 

Principle  of  science  applied,  22 

"  on  which  valid  arguments  are  constructed,  52. 

"  Value  of  a,  greater  as  it  is  more  simple,  54. 

•  Aristotle's  Dictum,  a  general,  55. 


368 


INDEX. 


Principle  the  same  in  the  ground  rules  for  simple  and  denominate 

numbers,  Sections  151-154,  232. 
"  of  science  and  rule  of  art,  179. 

Principles  should  be  separated  from  applications,  186,  187. 

of  science  are  general  truths,  208. 
"  of  Arithmetic,  how  taught,  208. 

should  precede  practice,  229. 

"  of  Mathematics,  deduced  from  definitions  and  axioms,  297 

Process  of  acquiring  mathematical  knowledge,  316-320. 

Product  of  several  numbers,  292. 

Progression,  Geometrical,  170. 

Property  of  a  figure,  256. 

PROPORTION The  relation  which  one  quantity  bears  to  another  with  re 
spect  to  its  being  greater  or  less,  163,  267-269 
"  Arithmetical  and  Geometrical,  163. 

Reciprocal  or  Inverse,  269. 
"  of  geometrical  figures,  270-273. 

PROPOSITION  A  judgment  expressed  in  words,  35. 

All  truth  and  all  error  lie  in  propositions,  also  answers  to 

all  questions,  36. 

"  formed  by  putting  together  two  names,  37. 

"  consists  of  three  parts,  38. 

subject,  and  predicate,  called  extremes,  38. 
"  Affirmative,  39  ;  Negative,  39. 

Three  propositions  essential  to  a  syllogism,  40. 
"         A  Universal,  62. 

"  Particular,  62. 


QUADRILATERAL....  A  portion  of  a  plane  bounded  by  four  straight  lines,  242. 
"       *  regarded  as  a  genus,  17. 

"  Different  varieties  of,  242. 

Quality  of  a  proposition  refers  to  its  being  affirmative  or  nega 

tive,  63. 

Quantities  only  of  the  same  kind  can  be  compared,  267. 

"  Two  classes  of,  in  Algebra,  287,  313. 

"  "         "         "  in  the  other  branches  of  Analysis,  282, 

283,  313. 
"  compared,  must  be  equal  or  unequal,  102,  307. 

QUANTITY Is  a  general  term  applicable  to  everything  which  can 

be  increased  or  diminished,  and  measured,  75,  321. 


INDEX. 


3Gi) 


Quantity,  Abstract,  does  not  involve  matter,  Sections  75,  96. 

"  Concrete  does,  75,  96, 

"  Propositions  divided  according  to,  62. 

"  presented  by  symbols,  93. 

"  consists  of  parts  which  can  be  numbered,  276 

"  Constant,  282. 

"  Variable,  282. 

"  Five  operations  can  be  performed  on,  288,  295. 

"  represented  by  five  signs,  289. 

"  Nature  of,  not  affected  by  the  sign,  290,  296. 

Questions  known,  when  all  propositions  are  known,  36. 

**  with  regard  to  number  and  space,  78. 

Analysis  of,  175,  176. 

u  Difficult,  in  Fractions  avoided,  191 

"  with  regard  to  methods  of  instruction,  371. 

Quotations  from  Kant,  21 ;  Sir  John  Herschel,  27,  322,  341,  359  ; 

Cousin,  180;  Olmsted's  Mechanics,  268;  Locke,  323  ; 
Mansfield's  Discourse  on  Mathematics,  325,  327  ;  Lord 
Bacon,  328  ;  Dr.  Barrow,  328,  340. 

Railways,  Problem  presented  in,  361. 

Rainbow,  Illustration,  322. 

RATIO -The  quotient  arising  from  dividing  one   number  or  quan 
tity  by  another,  163,  267. 

"  Discussion  concerning  it,  165-171. 

"  Arithmetical  and  Geometrical,  163. 

"  How  determined,  165. 

"  An  abstract  number,  267,  272. 

"  Terms  direct,  inverse,  or  reciprocal,  not  applicable  to,  269. 

Reading  in  Addition,  116,  117  ;  advantages  of,  118. 

in  Subtraction,  120. 
"  in  Multiplication,  122. 

«  in  Division,  123. 

"  of  figures,  its  aid  in  practical  operations,  230. 

Reason,  To  make  use  of  arguments,  42. 

"  A  premiss  placed  after  the  conclusion,  48. 

REASONING.. The  act  of  proceeding  from  certain  judgments  to  another, 

founded  on  them,  9. 

"  Three  operations  of  the  mind  concerned  in,  6. 

"  Process,  sameness  of  the,  42,  43,  45,  314. 

24 


370  INDEX. 


Reasoning  processes  of  mathematics  consist  of  two  parts,  Section  73. 

"  in  Analysis  is  based  on  the  supposition  that  we  are  deal 

ing  with  things,  2*78. 
Reciprocal  or          Inverse  Proportion,  269. 

RECTANGLE A  parallelogram  whose  angles  are  right  angles,  242. 

Remarks,  Concluding  subject  of  Arithmetic,  236. 

Reservoirs,  Croton,  description  of,  362. 
Right  angle  Definition  of,  258. 

Roman  Table,         when  taught,  215. 
Root,  Symbol  for  the  extraction  of,  295. 

Rule  of  Three,        Solution  of  questions  in,  169. 
Comparison  of  numbers,  186. 
"  should  precede  its  applications,  187. 

Rules,  Every  thing  done  according  to,  21. 

of  reasoning  analogous  to  those  of  Arithmetic,  45 
"  Advantages  of  logical,  50. 

"  for  teaching,  186. 

How  framed,  297. 

Scale  of  Tens,         Units  increasing  by,  124-130,  157,  183. 

SCIENCE In  its  popular  sense  means  knowledge  reduced  to  order, 

21,  326. 

"  In  its  technical  sense  means  an  analysis  of  the  laws  of 

nature,  21. 

"  contrasted  with  art,  22. 

"  of  Arithmetic,  172. 

"  Principles  of,  200,  208. 

"  Methods  of,  must  be  followed  in  Arithmetic,  228. 

of  Geometry,  237,  248,  257. 

"  Objects  and  means  of  pure,  322. 

"  should  be  made  as  much  deductive  as  possible,  336. 

"  Deductive  and  experimental,  337. 

"  when  experimental,  338,  339  ;  when  deductive,  338,  339. 

"  What  it  has  accomplished,  348. 

"  Practical  value  of,  in  factories,  358. 

"  "  "       "  in  constructing  steamships,  359. 

"  "  "      "  in  laying  out  and  measuring  land,  360. 

"       "  in  constructing  railways,  361. 

"  Its  power  illustrated  in  Croton  aqueduct,  362. 

"  What  constitutes  it,  372. 


INDEX.  371 


Second  Arithmetic,  its  place  and  construction,  Section  227-230. 
Sextant,  its  uses  in  Navigation,  359. 

SHADES,  SHADOWS,  AND  PERSPECTIVE — An  application  of  Descriptive  Geom 
etry,  381. 

SIGNIFICATE An  individual  for  which  a  common  term  stands,  15. 

Signs,  Five  used  to  denote  operations  on  quantity,  289. 

"  How  to  be  interpreted,  290. 

"  do  not  affect  the  nature  of  the  quantity,  290,  296. 

"  indicate  operations,  296,  298. 

SOLID A  portion  of  space  having  three  dimensions,  85. 

"  A  portion  of  space  combining  the  three  dimensions  of 

length,  breadth,  and  thickness,  246,  320 
"  Limit  of,  247. 

"  First  idea  of,  how  impressed,  320. 

Solids  bounded  by  plane  and  curved  surfaces,  85. 

"  Three  classes  of,  246. 

"  Analysis  of  comparison,  271,  272. 

"  Comparison  of,  under  the  supposition  of  changes  in  their 

volumes,  272. 

Solution  of  all  questions  in  the  Rule  of  Three,  169. 

«  of  an  equation  in  Algebra,  308. 

SPACE Is  indefinite  extension,  81,  82. 

"  has  three  dimensions,  length,  breadth,  and  thickness,  82 

"  Clear  conception  of,  necessary  to  understand  Geometry, 

238. 

SPECIES  One  of  the  divisions  of  a  genus  in  which  the  characteris 
tic  is  less  extensive,  but  more  full  and  complete,  16,  17. 
SUBSPECIES — One  of  the  divisions  of  a  species,  in  which 
the  characteristic  is  less  extensive,  but  more  full  and 
complete,  16,  19. 
LOWEST  SPECIES — A  species  which  cannot  be  regarded 

as  a  genus,  17. 
Spelling,  113;  in  Addition,  &c.,  115-123. 

SQUARE  A  quadrilateral  whose  sides  are  equal,  and  angles  right 

angles,  242. 
Statement  of  a  proposition  in  Algebra,  308. 

"  in  what  it  consists,  309. 

Steamship,  an  application  of  science,  359. 

SUBJECT The  name  denoting  the  person  or  thing  of  which  some 
thing  is  affirmed  or  denied,  38. 


372  INDEX. 


Subjects,  How  presented  in  a  text-book,  Section  209-212. 

Subtraction,  Readings  in,  120. 

Examples  in,  152. 
Combinations  in,  194. 

All  operations  in,  governed  by  one  principle,  232 
"  in  Algebra,  illustration  of,  298. 

Suggestions  for  teaching  Geometry,  273. 

for  teaching  Algebra,  315. 
Sum,  Its  definition,  203. 

SURFACE A  portion  of  space  having  two  dimensions,  84,  240,  319. 

"  Plane  and  Curved,  84,  240. 

Surfaces,  Curved,  245. 

"        of  Elementary  Geometry,  245. 
"  Limits  of,  247. 

SURVEYING The  application  of  the  principles  of  Trigonometry  to  the 

measurement  of  portions  of  the  earth's  surface,  379. 
A  branch  of  practical  science,  360. 

SYLLOGISM A   form    of    stating   the    connection  which    may   exist 

for  the  purpose  of  reasoning,  between  three  proposi 
tions,  40. 
"  A  formula  for  ascertaining  what  may  be  predicated. — 

How  it  accomplishes  this,  41. 
not  meant  by  Aristotle  to  be  the  form  in  which  arguments 

should  always  be  stated,  53. 
not  a  distinct  kind  of  argument,  54. 
"  an  argument  stated  at  full  length,  56. 

"  Symbols  used  for  the  terms  of,  56. 

Rules  for  examining  syllogisms,  67. 
"  has  three  and  only  three  terms,  67. 

"       "        "         "       "       propositions,  67. 
"  test  of  deductive  reasoning,  72,  99,  307. 

SYMBOLS The  letters  which  denote  quantities,  and  the  signs  which 

indicate  operations,  87,  93,  296. 
used  for  the  terms  of  a  syllogism,  56. 
Advantages  of,  57. 

Validity  of  the  argument  still  evident,  58. 
Truths  inferred  by  means  of,  true  of  all  things,  277. 
regarded  as  things  278. 
Two  classes  of,  in  analysis,  29& 
Abstract  and  concrete  quantity  represented  by,  321. 


INDEX.  373 


SYNTHESIS  The  process  of  first  considering  the  elements  separately, 

then  combining  them,  and  ascertaining  the  results  of 
combination,  Sections  89,  327. 

Synthetical  form,   for  what  best  adapted,  71,  89. 

Tables  of  Denominate  Numbers,  fractions  occur  five  times  in,  190. 

TECHNICAL Particular  and  limited  sense,  91. 

TERM Is  an  act  of  apprehension  expressed  in  words,  15. 

A  singular  term  denotes  but  a  single  individual,  15. 
"  A  common  denotes  any  individual  of  a  whole  class,  15. 

"         affords  the  means  of  classification,  16. 

Nature  of,  20. 

"         No  real  thing  corresponding  to,  20. 
"         Why  applicable  to  several  individuals,  20. 
MAJOR  TERM — The  predicate  of  the  conclusion,  40. 
MINOR  TERM — The  subject  of  the  conclusion,  40. 
MIDDLE  TERM — The  common  term  of  the  two  premises,  40. 
DISTRIBUTED — A  term  is  distributed  when  it  stands  for  all 

its  significates,  61. 

"  NOT  DISTRIBUTED — When  it  stands  for  a  part  of  its  sig 

nificates  only,  61. 

TERMS Two  of  the  three  parts  of  a  proposition,  38. 

"  The  antecedent  and  consequent  of  a  proportion,  164,  267. 

"  should  always  be  used  in  the  same  sense,  170,  205. 

TEXT-BOOK Should  be  an  aid  to  the  teacher  in  imparting  instruction, 

and  to  the  learner  in  acquiring  knowledge,  209. 

THICKNESS A  dimension  of  space,  82. 

Third  Arithmetic,  Principles   contained  in,   and  method    of   construction, 

231-236. 

Time,  Measure  of,  its  units  and  scale,  148. 
Topography,  Its  uses,  360. 

TRAPEZOID A  quadrilateral,  having  two  sides  parallel,  242. 

TRIANGLE A  portion  of  a  plane  bounded  by  three  straight  lines,  241. 

"  The  simplest  plane  figure,  241. 

Different  kinds  of,  241. 
*'  regarded  as  a  genus,  256. 

TRIGONOMETRY  ....An  application  of  the  principles  of  Arithmetic,  Algebra, 
and  Geometry  to  the  determination  of  the  sides  and 
angles  of  triangles,  378. 
"  Plane  and  Spherical,  378. 


374  INDEX. 


Troy  Weight,          Its  units  and  scale,  Section  137. 

TRUTH An  exact  accordance  with  what  has  been,  is,  or  shall 

be,  24. 

Two  methods  of  ascertaining,  24. 
is  inference  from  facts  or  other  truths,  24,  25. 
"  regarded  as  a  species,  25. 

How  inferred  from  facts,  26. 
"  A  true  proposition,  36. 

TRUTHS  INTUITIVE  OR  SELF-EVIDENT — Are  such  as  become  known 

by  considering  all  the  facts  on  which  they  depend,  an  1 
apprehending  the  relations  of  those  facts  at  the  s::me 
time,  and  by  the  same  act  by  which  we  apprehend  the 
facts  themselves,  27. 
"  LOGICAL — Those  inferred  from  numerous  and  complicated 

facts ;  and  also,  truths  inferred  from  truths,  28. 
"  of  Geometry,  237. 

"  Three  classes  of,  237. 

"  Demonstrative,  237. 

Unit  fixed  by  the  place  of  the  figure,  127. 

"  of  the  fraction,  160,  161. 

"  of  the  expression,  160. 

Unities  Advantages  of  the  system  of,  150-154. 

UNIT  OF  MEASURE  .  .The  standard  for  measurement,  94. 
"  for  lines,  surfaces,  solids,  249. 

"  only  basis  for  estimating  quantity,  251. 

UNIT  ONE A  single  thing,  104. 

"  All  numbers  come  from,  108,  109, 132, 150. 

Method  of  impressing  its  values,  133. 

"  Three  kinds  of  operations  performed  upon,  182-186. 

Units,  Abstract  or  simple,  111,  132. 

"  Denominate  or  Concrete,  111. 

"  of  currency,  132. 

"  of  weight,  132. 

"  of  measure,  132,  139,  249. 

of  length,  140. 
"  of  surface,  141. 

Duodecimal,  142. 
"  of  solidity,' 145. 

"  Fractional,  155,  185. 


INDEX. 


375 


UNTTY — UNIT Any  thing  regarded  as  a  whole,  Sections  109,  110 

Universal  Proposition,  62. 

Utility  and  Progress,  leading  ideas,  page  11. 

VARIABLES  Quantities  which  undergo  certain  changes  of  value,  the 

laws  of  which  are  indicated  by  the  algebraic  expres 
sions  into  which  they  enter,  282,  283,  313. 
"  represented  by  the  final  letters  of  the  alphabet,  284. 

Variations,  Theory  of,  285. 

Varying  Scales,      Units  increasing  by,  131,  183. 

Velocity  known  by  measurement,  95. 

Weight  known  by  measurement,  95. 

"  A,  should  be  exhibited  to  give  ideas  of  numbers,  133. 

"  Standard  for,  254. 

Whateley,  Archbishop,  his  views  of  logic,  72. 

Words,  Definition  of,  113. 

"  expressing  results  of  combinations,  193-197. 

"  Double  or  incomplete  sense  of,  322. 


ZERO  The  limit  of  a  decreasing  quantity,  302-306- 


THE    END. 


A.  S.  BARNES  t  COMPANY  S  PUBLICATIONS. 
Davits'  System  of  Mathematics. 

MATHEMATICAL  WORKS, 

IN  A   SERIES   OF   THREE   PARTS! 

ARITHMETICAL,  ACADEMICAL,  AND  COLLEGIATE. 

BY  CHARLES  DAYIES,  Lt.I) 


I.    THE  ARITHMETICAL  COURSE  FOR  SCHOOLS. 

1.  PRIMARY  TABLE-BOOK. 

2.  FIRST  LESSONS  IN  ARITHMETIC. 

3.  SCHOOL  ARITHMETIC.     (Key  separate.) 

4.  GRAMMAR  OF  ARITHMETIC. 

II.    THE  ACADEMIC  COURSE. 

1.  THE  UNIVERSITY  ARITHMETIC.     (Key  separate.) 

2.  PRACTICAL  GEOMETRY  AND  MENSURATION. 

3.  ELEMENTARY  ALGEBRA.     (Key  separate.) 

4.  ELEMENTARY  GEOMETRY. 

5.  ELEMENTS  OF  SURVEYING. 

III.    THE  COLLEGIATE  COURSE. 

1.  DAVIES'  BOURDON'S  ALGEBRA. 

2.  DAVIES'  LEGENDRE'S  GEOMETRY  AND  TRIGONOMETRY. 

3.  DAVIES'  ANALYTICAL  GEOMETRY. 

4.  DAVIES'  DESCRIPTIVE  GEOMETRY. 

5.  DAVIES'  SHADES,  SHADOWS,  AND  PERSPECTIVE. 

6.  DAVIES'  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

.   DAVIES'  LOGIC  AND  UTILITY  OF  MATHEMATICS. 

This  series,  combining  all  that  is  most  valuable  in  the  various  methods  of  European 
instruction,  impcoved  and  matur«d  by  the  suggestions  of  more  than  thirty  years'  expe 
rience,  now  forms  the  only  complete  consecutive  course  of  Mathematics.  It*  methods, 
narmsnizir.g  as  the  works  of  one  mind,  carry  the  student  onward  by  the  same  analogies 
uid  the  same  laws  of  association,  and  are  calculated  to  impart  a  comprehensive  knowl 
edge  of  the  science,  combining  clearness  in  the  several  branches,  and  unity  and  propor 
tion  in  the  whole.  Being  the  system  so  long  in  use  at  West  Point,  through  which  so 
many  men,  eminent  for  their  scientific  attainments,  have  passed,  and  hf  ing  been 
adopted,  as  Text  Books,  by  most  of  the  colleges  in  the  United  States,  it  may  be  justly 
regarded  as  our 

NATIONAL  SYSTEM  OF  MATHEMATICS. 


A.  S.  BARNES  &   COMPANY'S  PUBLICATIONS. 

Chambers'  Educational   Court*  . 

CHAMBERS'  EDUCATIONAL  COURSE. 

THE    SCIENTIFIC    SECTION, 


The  Messrs.  Chambers  have  employed  the  first  professors  in  Scotland  in  the  prepara 
tion  of  these  works.  They  are  now  offered  to  the  schools  of  the  United  States,  under 
the  American  revision  of  D.  M.  REESE,  M.D.,  LL.D.,  late  Superintendent  of  Public 
Schools  in  the  city  and  county  of  New  York. 

I.  CHAMBERS'  TREASURY  OF  KNOWLEDGE, 

II.  CLARK'S  ELEMENTS  OF  DRAWING  AND  PERSPECTIVE. 

III.  CHAMBERS'  ELEMENTS  OF  NATURAL  PHILOSOPHY. 

IV.  REID  &  BAIN'S  CHEMISTRY  AND  ELECTRICITY. 

V.    HAMILTON'S  VEGETABLE  AND  ANIMAL  PHYSIOLOGY 
VI.    CHAMBERS'  ELEMENTS  OF  ZOOLOGY. 
VII.    PAGE'S  ELEMENTS  OF  GEOLOGY. 


"It  is  well  known  that  the  original  publishers  of  these  works  (the  Messrs.  Chambers 
of  Edinburgh)  are  able  to  command  the  best  talent  in  the  preparation  of  their  books, 
and  that  it  is  their  practice  to  deal  faithfully  with  the  public.  This  series  will  not  dis 
appoint  the  reasonable  expectations  thus  excited.  They  are  elementary  works  pre 
pared  by  authors  in  every  way  capable  of  doing  justice  to  their  respective  undertakings, 
and  who  have  evidently  bestowed  upon  them  the  necessary  time  and  labor  to  adapt 
them  to  their  purpose.  We  recommend  them  to  teachers  and  parents  with  confidence. 
If  not  introduced  as  class-books  in  the  school,  they  may  be  used  to  excellent  advantage 
in  gemral  exercises,  and  occasional  class  exercises,  for  which  every  teacher  ought  to 
provide  himself  with  an  ample  store  of  materials.  The  volumes  may  be  had  separate 
ly  ;  and  the  one  first  named,  in  the  hands  of  a  teacher  of  the  younger  classes,  might 
furnish  an  inexhaustible  fund  of  amusement  and  instruction.  Together,  they  would 
constitute  a  rich  treasure  to  a  family  of  intelligent  children,  and  impart  a  thirst  for 
knowledge." — Vermont  Chronicle. 

"Of  all  the  numerous  works  of  this  class  that  have  been  published,  there  are  none 
that  have  acquired  a  more  thoroughly  deserved  and  high  reputation  than  this  series. 
The  Chambers,  of  Edinburgh,  well  known  as  the  careful  and  intelligent  publishers  of 
a  vast  number  of  works  of  much  importance  in  the  educational  world,  .ore  the  fathers 
of  thi&  series  of  books,  and  the  American  editor  has  exercised  an  unusual  degree  of 
judgment  in  their  preparation  for  the  use  of  schools  as  well  as  private  families  in  this 
country." — Philad.  Bulletin. 

'The  titles  furnish  a  key  to  the  contents,  and  it  is  only  necessary  for  us  to  say,  that 


the  malerial  of  each  volume  is  admirably  worked  up,  presenting  with  sufficient  fulnew 
and  wilh  much  clearness  of  method  the  several  subjects  which  are  treated."— Ctn 
Gazette. 


u  We  notice  these  works,  not  merely  because  they  are  school  books,  but  for  the  pur 
pose  of  expressing  our  thanks,  as  the  'advocate'  of  the  educational  interenU  of  the 
people  and  their  children,  to  the  enterprising  publishers  of  these  and  many  other  val 
uable  works  of  the  same  character,  the  tendency  of  which  is  to  diffuse  useful  know 
ledge  throughout  the  masses,  for  the  good  work  they  are  doing,  and  the  hope  thai 
thoir  reward  may  be  commensurate  with  their  deserts,"— Maine  School  jtdvvnati. 


A.  S.  BARNES  <k  COMPANY  S  PUBLICATIONS. 
Parker's    Natural  Philosophy. 

NATURAL  AND  EXPERIMENTAL  PHILOSOPHY, 

FOR  SCHOOLS  AND  ACADEMIES. 

BY  R.  G.  PARKER,  A.  M., 

.*«£/»or  of  u  Rhetorical  Reader,"   "Exercises  in  English  Composition,"  "Oullinet 
of  History,"  etc.,  etc. 


I.  PARKER'S  JUVENILE  PHILOSOPHY. 

II.  PARKER'S  FIRST  LESSONS  IN  NATURAL  PHILOSOPHY. 
III.  PARKER'S  SCHOOL  COMPENDIUM  OF  PHILOSOPHY. 

The  use  of  school  apparatus  for  illustrating  and  exemplifying  the  principles  of  Natural 
and  Experimental  Philosophy,  has,  within  the  last  few  years,  become  so  general  as 
to  render  necessary  a  work  which  should  combine,  in  the  same  course  of  instruction, 
the  theory,  with  a  full  description  of  the  apparatus  necessary  for  illustration  and 
experiment.  The  work  of  Professor  Parker,  it  is  confidently  believed,  fully  meets  that 
requirement.  It  is  also  very  full  in  the  general  facts  which  it  presents — clear  and 
concise  in  its  style — and  entirely  scientific  and  natural  in  its  arrangement. 

"  This  work  is  better  adapted  to  the  present  state  of  natural  science  than  any  other 
similar  production  with  which  we  are  acquainted." —  Wayne  Co.  Whig. 

"•  This  is  a  school-book  of  no  mean  pretensions  and  no  ordinary  value." — Albany 
Spectator. 

"  We  predict  for  this  valuable  and  beautifully-printed  work  the  utmost  success." — 
Newark  Daily  Advertiser. 

"  The  present  volume  strikes  us  as  having  very  marked  merit."— JV.  Y.  Courier. 

"  It  seems  to  me  to  have  hit  a  happy  medium  between  the  too  simple  and  the  two 
abstract." — B.  A.  Smith,  Principal  of  Leicester  Academy,  Mass. 

"  I  have  no  hesitation  in  saying  that  Parker's  Natural  Philosophy  is  the  most  valuable 
elementary  work  I  have  seen." — Gilbert  Langdon  Hmne,  Prof.  JVat.  Phil.  JV.  Y.  City. 

*•'  I  am  happy  to  say  that  Parker's  Philosophy  will  be  introduced  and  adopted  in 
'  Victoria  College,'  at  the  commencement  of  the  next  collegiate  year  in  autumn  ;  and  I 
hope  that  will  be  but  the  commencement  of  the  use  of  so  valuable  an  elementary  work 
in  our  schools  in  this  country.  The  small  work  of  Parker's  (Parker's  First  Lessons)  was 
introduced  the  last  term  in  a  primary  class  of  the  institution  referred  to,  and  that  with 
great  success.  I  intend  to  recommend  its  use  shortly  into  the  model  school  in  this  city, 
uud  the  larger  work  to  the  students  of  the  provincial  Normal  School." — E.  Ryerson, 
Superintendent  of  Public  Instruction  of  Upper  Canada. 

"  I  have  examined  Parker's  First  Lessons  and  Compendium  of  Natural  and  Expert- 
mental  Philosophy,  and  am  much  pleased  with  them.  I  have  long  felt  dissatisfaction 
with  the  Text-Books  on  this  subject  most  in  use  in  this  section,  and  am  happy  now  to 
find  books  that  I  can  recommend.  I  shall  introduce  them  immediately  into  my  school." 
Hiram  Orcutt,  Principal  of  Tlietford  Academy,  Vermont. 

"  I  have  no  hesitation  in  pronouncing  it  the  best  work  on  the  subject  now  published, 
We  shall  use  it  here,  and  I  have  already  secured  its  adoption  in  some  of  the  high- 
schools  and  academies  in  our  vicinity." — M.  D.  Leggett,  Sup.  of  Warren  Public  Schools. 

"  We  are  glad  to  see  this  little  work  on  natural  philosophy,  because  the  amount  of 
valuable  information  under  all  these  heads,  to  be  gained  from  it  by  any  little  boy  or 
girl,  is  Inestimable.  It  puts  them,  too,  upon  the  right  track  after  knowledge,  and  pre 
vents  tbeir  minds  from  being  weakened  and  wasted  by  the  sickly  sentimentality  of 
tales,  novels,  and  poetry,  which  will  always  occupy  the  attention  of  the  mind  wh«n 
w/hing  more  useful  has  taken  possession  of  it." — Mississippian. 


A.   S.   BARNES  &  COMPANY  S  PUBLIC  .TIONS. 
Willard's  School  Histories  and  Charts. 

MRS.  EMMA  WILLARD'S 

SERIES  OF  SCHOOL  HISTORIES  AND  CHARTS. 


I.  WILLARD'S  UNIVERSAL  HISTORY  IN  PERSPECTIVE.    $1.50. 

II.  WILLARD'S  TEMPLE  OF  TIME.    Mounted,  $1.25.    Bound,  75  eta 

III.  WILLARD'S  HISTORIC  GUIDE.    SOcts. 

IV.  WILLARD'S  ENGLISH  CHRONOGRAPHER, 

WILLARD'S  UNITED  STATES. 

The  Hon.  Dan.  Webster  says,  of  an  early  edition  of  the  above  work,  in  a  letter  to 
the  author,  "I  KKKP  IT  NEAR  ME,  AS  A  BOOK  OF  REFERENCE,  ACCURATE  IN  FACTS  AND 

DATES  " 

"THE  COMMITTEE  ON  BOOKS  OF  THE  WARD  SCHOOL  ASSOCIATION  RESPECTFULLY 
REPORT : 

"That  they  have  examined  Mrs.  Willard's  History  of  the  United  States  with  peculiar 
interest,  and  are  free  to  say,  that  it  is  in  their  opinion  decidedly  the  best  treatise  on 
this  interesting  subject  that  they  have  seen.  As  a  school-book,  its  proper  place  is 
among  the  first.  The  language  is  remarkable  for  simplicity,  perspicuity,  and  neatness ; 
youth  could  not  be  trained  to  a  better  taste  for  language  than  this  is  calculated  to  im 
part.  It  places  at  once,  in  the  hands  of  American  youth,  the  history  of  their  country 
from  the  day  of  its  discovery  to  the  present  time,  and  exhibits  a  clear  arrangement  of 
all  the  great  and  good  deeds  of  their  ancestors,  of  which  they  now  enjoy  the  benefits, 
and  inherit  the  renown.  The  struggles,  sufferings,  firmness,  and  piety  of  the  first  settlers 
are  delineated  with  a  masterly  hand." — Extract  from  a  Report  of  the  Ward  School 
Teachers'  Association  of  the  City  of  New  York. 

"  We  consider  the  work  «  remarkable  one,  in  that  tt  forms  the  best  book  for  general 
reading  and  reference  published,  and  at  the  same  time  has  no  equal,  in  our  opinion,  as 
a  text-book.  On  this  latter  point,  the  profession  which  its  author  has  so  long  followed 
with  such  signal  success,  rendered  her  peculiarly  a  fitting  person  to  prepare  a  text 
book." — Boston  Traveller. 


u  MRS.  WILLARD'S  SCHOOL  HISTORY  OF  THE  UNITED  STATES. — It  is  one  of  those 
rare  things,  a  good  school-book ;  infinitely  better  than  any  of  the  United  Sla»es  Historww 
fitted  for  schools,  which  we  have  at  present." — Cincinnati  Gazette. 

*' We  think  we  are  warranted  in  saying,  that  it  is  better  adapted  to  meet  the  wants 
of  our  schools  and  academies  in  which  history  is  pursued,  than  any  other  work  of  the 
kind  now  before  the  public.  The  style  is  perspicuous  and  flowing,  and  the  prominent 
points  of  our  history  are  presented  in  such  a  manner  as  to  make  a  deep  and  lasting 
impression  on  the  mind.  We  could  conscientiously  say  much  more  in  praise  of  this 
book,  but  must  content  ourselves  by  heartily  commending  it  to  the  attention  of  those 
who  are  anxious  to  find  a  gc  od  text-book  of  American  history  for  the  use  of  schools."— 
tfevburyport  Watchman. 


I.    WILLARD'S    HISTORY  OF  THE   UNITED   STATES,  OR   RE 
PUBLIC  OF  AMERICA.    8vo.    Price  $1.50. 
II.    WILLARD'S  SCHOOL   HISTORY  OF  THE   UNITED  STATES. 

63  cts. 
III.    WILLARD'S  AMERICAN  CHRONOGRAPHER.    $1.50. 


A.  S.   BARNES   Sc.  COMPANY  S  PUBLICATIONS. 
Parker's  Rhetorical  Reader. 


PARKER'S  RHETORICAL  READER.    12mo. 

Exercises  in  Rhetorical  Reading,  designed  to  familiarize  readers  with  the 
pauses  and  other  marks  in  general  use,  and  lead  them  to  the  practice  of 
modulation  and  inflection  of  the  voice.  By  R.  G.  PARKER,  author  of  "  Ex 
ercises  in  English  Composition,"  "  Compendium  of  Natural  Philosophy," 
&c.,  &c. 

This  work  possesses  many  advantages  which  commend  it  to  favor,  among  which  ar* 
the  following:— It  is  adapted  to  all  classes  and  schools,  from  the  highest  to  the  lowest. 
It  contains  a  practical  illustration  of  all  the  marks  employed  in  written  language : 
also  lessons  for  the  cultivation,  improvement,  and  strengthening  of  the  voice,  and 
instructions  as  well  as  exercises  in  a  great  variety  of  the  principles  of  Rhetorical 
Reading,  which  cannot  fail  to  render  it  a  valuable  auxiliary  in  the  hands  of  any 
teacher.  Many  of  the  exercises  are  of  sufficient  length  to  afford  an  opportunity  for 
each  member  of  any  class,  however  numerous,  to  participate  in  the  same  exercise — a 
feature  which  renders  it  convenient  to  examining  committees.  The  selections  for 
exercises  in  reading  are  from  the  most  approved  sources,  possessing  a  salutary  moral 
and  religious  tone,  without  the  slightest  tincture  of  sectarianism. 


"  I  have  to  acknowledge  the  reception  through  your  kindness  of  several  volumes.  1 
have  not  as  yet  found  time  to  examine  minutely  all  the  books.  Of  Mr.  Parker's  Rhe 
torical  Reader,  however,  I  am  prepared  to  speak  in  the  highest  terms.  I  think  it  so 
well  adapted  to  the  wants  of  pupils,  that  I  shall  introduce  it  immediately  in  the  Acad 
emy  of  which  I  am  about  to  take  charge  at  Madison,  in  this  state.  It  is  the  best  thing 
of  the  kind  I  have  yet  found.  I  cannot  say  too  much  in  its  favor."— John  O.  Clark* 
Rector  of  the  Madison  Male  Academy,  Athens,  Oa. 


"  Mr.  Parker  has  made  the  public  his  debtor  by  some  of  his  former  publications— 
especially  the 'Aids  to  English  Composition' — and  by  this  he  has  greatly  increased 
the  obligation.  There  are  reading  books  almost  without  number,  but  very  lew  of 
them  pretend  to  give  instructions  how  to  read,  and,  unluckily,  few  of  our  teachers  are 
competent  to  supply  the  defect.  If  young  persons  are  to  be  taught  to  read  well,  il 
must  generally  be  done  in  the  primary  schools,  as  the  collegiate  term  affords  too  little 
time  to  begin  and  accomplish  that  work.  We  have  seen  no  other  'Reader'  with 
which  we  have  been  so  well  pleased ;  and  as  an  evidence  of  our  appreciation  of  its 
worth,  we  shall  lay  it  aside  for  the  use  of  a  certain  juvenile  specitieu  of  humanity  in 
wh0se  affairs  we  are  specially  interested." — Christian  Advocate. 

"  We  cannot  too  often  urge  upon  teachers  the  importance  of  rsading,  as  a  part  of 
education,  and  we  regard  it  as  among  the  auspicious  signs  of  the  times,  that  so  much 
more  attention  is  given,  by  the  best  of  teachers,  to  the  cultivation  of  a  power  which  ia 
at  once  a  most  delightful  accomplishment,  and  of  the  first  importance  as  a  means  of 
discipline  and  progress.  In  this  work,  Mr.  Parker's  volume,  we  are  sure,  will  be  found 
a  valuable  aid." — Vermont  Chronicle. 

"The  title  of  this  work  explains  its  character  and  design,  which  are  well  carried  out 
by  the  manner  in  which  it  is  executed.  As  a  class-book  for  students  in  elocution,  or  as 
an  ordinary  reading  book,  we  do  not  think  we  have  seen  any  thing  superior.  The  dis 
tinguishing  characteristic  of  its  plan  is  to  assume  some  simple  and  i.amiliar  example, 
which  will  be  readily  understood  by  the  pupil,  and  which  Nature  will  tell  him  how  to 
deliver  properly,  and  refer  more  difficult  passages  to  this,  as  a  model.  There  is,  how 
ever,  another  excellence  in  the  work,  which  we  take  pleasure  in  commending;  it  is 
the  progressiveness  with  which  the  introductory  lessons  are  arranged.  In  teaching 
every  art  and  science  this  is  indispensable,  and  in  none  more  so  than  in  that  of  elocu 
tion.  The  pieces  for  exercise  in  reading  are  selected  with  much  taste  and  judgment 
We  have  no  doubt  that  those  who  MSB  this  book  will  be  satisfied  with  its  nicceM."— 
Teach-***  Advocate. 

q 


A.  S.  BARNES   &   COMPANY  S  PUBLICATIONS. 
Brooks's   Greek  and  Latin   Classics. 

PROFESSOR  BROOKS'S 
GEEEK  AND  LATIN  CLASSICS. 


THIS  series  of  the  GREEK  and  LATIN  CLASSICS,  by  N.  C.  Brooks,  of  Baltimore,  is  on  an 
improved  plan,  with  peculiar  adaptation  to  the  wants  of  the  American  student.  To 
secure  accuracy  of  text  in  the  works  that  are  to  appear,  the  latest  and  most  approved 
European  editions  of  the  different  classical  authors  will  be  consulted.  Original  illus 
trative  and  explanatory  notes,  prepared  by  the  Editor,  will  accompany  the  text. 
These  notes,  though  copious,  will  be  intended  to  direct  and  assist  the  student  in  his 
labors,  rather  than  by  rendering  every  thing  too  simple,  to  supersede  the  necessity  of 
due  exertion  on  his  own  part,  and  thus  induce  indolent  habits  of  study  and  reflection, 
and  feebleness  of  intellect. 

In  the  notes  that  accompany  the  text,  care  will  be  taken,  on  all  proper  occasions,  to 
develop  and  promote  in  the  mind  of  the  student,  sound  principles  of  Criticism, 
Rhetoric,  History,  Political  Science,  Morals,  and  general  Religion;  so  that  he  may  con 
template  the  subject  of  the  author  he  is  reading,  not  within  the  circumscribed  limits 
of  a  mere  rendering  of  the  text,  but  consider  it  in  all  its  extended  connections — and 
thus  learn  to  think,  as  well  as  to  translate. 

BROOKS'S  FIRST  LATIN  LESSONS. 

This  is  adapted  to  any  Grammar  of  the  language.  It  consists  of  a  Grammar,  Reader, 
tud  Dictionary  combined,  and  will  enable  any  one  to  acquire  a  knowledge  of  the  ele 
ments  of  the  Latin  Language,  without  an  instructor.  It  has  already  passed  through 
five  editions.  18mo. 

BROOKS'S  C/ESAR'S  COMMENTARIES.    (In  press.) 
This  edition  of  the  Commentaries  of  Caesar  on  the  Gallic  War,  besides  critical  and 
explanatory  notes  embodying  much  information,  of  an  historical,  topographical,  and 
military  character,  is  illustrated  by  maps,  portraits,  views,  plans  of  battles,  &c.    It  has 
a  good  Clavis,  containing  all  the  words.    Nearly  ready.    12mo. 

BROOKS'S  OVID'S  METAMORPHOSES.    8vo. 

This  edition  of  Ovid  is  expurgated,  and  treed  from  objectionable  matter.  It  is  eluci 
dated  by  an  analysis  and  explanation  of  the  fables,  together  with  original  English  notes, 
historical,  mythological,  and  critical,  and  illustrated  by  pictorial  embellishments  ;  with 
a  Clavis  giving  the  meaning  of  all  the  words  with  critical  exactness.  Each  fable  con 
tains  a  plate  from  an  original  design,  and  an  illuminated  initial  letter. 

BROOKS'S  ECLOGUES  AND  GEORGICS  OF  VIRGIL.    (In  press.) 
This  edition  of  Virgil  is  elucidated  by  copious  original  notes,  and  extracts  from 

ancient  and  modern  pastoral  poetry.    It  is  illustrated  by  plates  from  original  designs, 

and  contains  a  Clavis  giving  the  meaning  of  all  the  words.    8vo. 

BROOKS'S  FIRST  GREEK  LESSONS.    12mo. 

This  Greek  elementiiry  is  on  the  same  plan  as  the  Latin  Lessons,  and  affords  equal 
facilities  to  the  student.  The  paradigm  of  the  Greek  verb  has  been  greatly  simplified 
anl  valuable  exercises  in  comparative  philology  introduced. 

BROOKS'S  GREEK  COLLECTANEA  EVANGELICA.    12mo. 
This  consists  of  portions  of  the  Four  Gospels  in  Greek,  arranged  in  Chronological 
order ;  and  forms  a  connected  history  of  the  principal  events  in  the  Saviour's  life  and 
ministry.    It  contains  a  Lexicon,  and  is  illustrated  and  explained  by  notes. 

BROOKS'S  GREEK  PASTORAL  POETS.    (In press.) 
This  contains  the  Greek  Idyls  of  Theocritus,  Bion,  and  Moschus,  elucidated  by  note* 
and  copious  extracts  from  ancient  and  modern  pastoral  poetry.    Each  Idyl  is  illustrated 
by  beautiful  plates  from  original  designs.    It  contains  a  good  Lexicon. 


\.  S.  BARNES  &   COMPANY  S  PUBLICATIONS. 
Page's  Theory  and  Practice  of  Teaching. 

THEORY  AND  PRACTICE  OF  TEACHING \ 

OR   THE 

MOTIVES    OF    GOOD    SCHOOL-KEEPING. 

BY  DAVID  PAGE,  A.M., 

tATI  PRINCIPAL  OF  THE  STATE  NORMAL  SCHOOL,  NEW  YORK. 


"  I  received  a  few  days  since  your  '  Theory  and  Practice,  &c.,'  and  a  capital  theory 
End  capital  practice  it  is.  I  have  read  it  with  immingled  delight.  Even  if  I  should 
look  through  a  critic's  microscope,  I  should  hardly  find  a  single  sentiment  to  dissent 
from,  and  certainly  not  one  to  condemn.  The  chapters  on  Prizes  and  on  Corporal 
Punishment  are  truly  admirable.  They  will  exert  a  most  salutary  influence.  So  of  the 
views  sparsim  on  moral  and  religious  instruction,  which  you  so  earnestly  and  feelingly 
insist  upon,  and  yet  within  true  Protestant  limits.  IT  is -A  GRAND  BOOK,  AND  I  THANK 
HEAVEN  THAT  YOU  HAVE  WRITTEN  IT." — Hon.  Horace  Mann,  Secretary  of  the  Board  of 
Education  in  Massachusetts. 


u  Were  it  our  business  to  examine  teachers,  we  would  never  dismiss  a  candidate 
without  naming  this  book.  Other  things  being  equal,  we  would  greatly  prefer  a  teacher 
who  has  read  it  and  speaks  of  it  with  enthusiasm.  In  one  indifferent  to  such  a  work, 
we  should  certainly  have  little  confidence,  however  he  might  appear  in  other  respects. 
Would  that  every  teacher  employed  in  Vermont  this  winter  had  the  spirit  of  this  book 
in  his  bosom,  its  lessons  impressed  upon  his  heart!" — Vermont  Chronicle. 


"I  am  pleased  with  and  commend  this  work  to  the  attention  of  school  teachers,  and 
those  who  intend  to  embrace  that  most  estimable  profession,  for  light  and  instruction 
to  guide  and  govern  them  in  the  discharge  of  their  delicate  and  important  duties." — 
JV.  S.  Benton,  Superintendent  of  Common  Schools,  State  of  New  York. 


Hon.  S.  Young-  says,  "It  ia  altogether  the  best  book  on  this  subject  1  have  erer 
seen." 


President  North,  of  Hamilton  College,  says,  "  I  have  read  it  with  all  that  absorbing 
self-denying  interest,  which  in  my  younger  days  was  reserved  for  fiction  and  poetry.  I 
am  delighted  with  the  book." 

Hon.  Marcus  S.  Reynolds  says,  "  It  will  do  great  good  by  showing  the  Teacher  what 
should  be  his  qualifications,  and  what  may  justly  be  required  and  expected  of  him." 


"I  wish  you  would  send  an  agent  through  the  several  towns  of  this  State  with 
Page's  'Theory  and  Practice  of  Teaching,'  or  take  some  other  way  of  bringing  this 
valuable  book  to  the  notice  of  every  family  and  of  every  teacher.  I  should  be  rejoiced 
to  see  the  principles  which  it  presents  as  to  the  motives  and  methods  of  good  school- 
keeping  carried  ut  in  every  school-room ;  and  as  nearly  as  possible,  in  the  style  in 
which  Mr.  Page  illustrates  them  in  his  own  practice,  as  the  devoted  and  accomplished 
Principal  of  your  State  Normal  School." — Heni~y  Barnard,  Superintendent  of  Common 
Schools  for  the  State  of  Rhode  Island. 

"The  'Theory  and  Practice  of  Teaching,'  by  D.  P.  Page,  is  one  of  the  best  books  of 
the  kind  1  have  ever  met  with.  In  it  the  theory  and  practice  of  the  teacher's  duties 
are  clearly  explained  and  happily  combined.  The  style  is  easy  and  familiar,  and  the 
suggestions  it  contains  are  plai-n,  practical,  and  to  the  point.  To  teachers  especially  it 
will  furnish  very  important  aid  in  discharging  the  duties  of  their  high  and  responsible 
profession."— Roger  S.  Hoioard,  Superintendent  of  Common  Schools,  Orangt  Co.,  Vi. 


A.  S.   BARNES  A;  COMPANY'S  PUBLICATIONS. 


Science  of  the  English  Language 


CLARK'S  NEW  ENGLISH  GRAMMAR. 

Practical  Grammar,  in  which  WORDS,  PHRASES,  and  SENTENCES  are  classi 
fied,  according  to  their  offices,  and  their  relation  to  each  other:  illustrated 
by  a  complete  system  of  Diagrams.  By  S.  W.  CLARK,  A.  M.  Price  50  cts. 


"It  is  a  most  capital  work,  and  well  calculated,  if  we  mistake  not,  to  supersede,  even 
in  our  best  schools,  works  of  much  loftier  pretension.  The  peculiarity  of  ita  method 
grew  out  of  the  best  practice  of  its  author  (as  he  himself  assures  us  in  its  preface) 
while  engaged  in  communicating  the  science  to  an  adult  class ;  and  his  success  was 
fully  commensurate  with  the  happy  and  philosophic  design  he  has  unfolded." — Rahway 
Register. 

"This  new  work  strikes  us  very  favorably.  Its  deviations  from  older  books  of  the 
kind  are  generally  judicious  and  often  important.  We  wish  teachers  would  examine 
it."— JVcto  York  Tribune. 


"  It  is  prepared  upon  a  new  plan,  to  meet  difficulties  which  the  author  has  encoun 
tered  in  practical  instruction.  Grammar  and  the  structure  of  language  are  taught 
throughout  by  analysis,  and  in  a  way  which  renders  their  acquisition  easy  and  satisfac 
tory.  From  the  slight  examination,  which  is  all  we  have  been  able  to  give  it,  we  are 
convinced  it  has  points  of  very  decided  superiority  over  any  of  the  elementary  works 
in  common  use.  We  commend  it  to  the  attention  of  all  who  are  engaged  in  instruc 
tion." — New  York  Courier  and  Enquirer. 

"  From  a  thorough  examination  of  your  method  of  teaching  the  English  language, 
I  am  prepared  to  give  it  my  unqualified  approbation.  It  is  a  plan  original  and  beau 
tiful — well  adapted  to  the  capacities  of  learners  of  every  age  and  stage  of  advance 
ment." — A.  R.  Simmons,  Ex-Superintendent  of  Bristol. 

"I  have,  under  my  immediate  instruction  in  English  Grammar,  a  class  of  more  than 
fifty  ladies  and  gentlemen  from  the  Teachers'  Department,  who,  having  studied  the 
grammars  in  common  use,  concur  with  me  in  expressing  a  decided  preference  for 
'Clark's  New  Grammar,'  which  we  have  used  as  a  text-book  since  its  publication,  and 
which  will  be  retained  as  such  in  this  school  hereafter." — Professor  Brittan,  Principal 
of  Lyons  Union  School. 

"Clark's  Grammar  I  have  never  seen  equalled  for  practicability,  which  is  of  the  ut 
most  importance  in  all  school-books." — 5.  B.  Clark,  Principal  of  Scarborough  Jlcadr 
emy,  Maine. 

"The  Grammar  is  just  such  a  book  as  I  wanted,  and  I  shall  make  it  the  text-book  in 
my  school." —  William  Brickley,  Teacher  at  Canastota,  JV*.  Y. 

"This  original  production  will,  doubtless,  become  an  indispensable  auxiliary  to  re 
store  the  English  language  to  its  appropriate  rank  in  our  systems  of  education.  After 
a  cursory  perusal  of  its  contents,  we  are  tempted  to  assert  that  it  foretells  the  dawn  of 
a  brighter  age  to  our  mother  tongue." — Southern,  Literary  Gazette. 

"  I  have  examined  your  work  on  Grammar,  and  do  not  hesitate  to  pronounce  it  su 
perior  to  any  work  with  which  I  am  acquainted.  I  shall  introduce  it  into  the  Mount 
Morris  Union  School  at  the  first  proper  opportunity." — //.  O.  Winslow,  Jl.  J\L,  Princi 
pal  of  Mnunt  Morris  Union  School. 

"Professor  Clark's  new  work  on  Grammar,  containing  Diagrams  illustrative  of  his 
system,  is,  in  my  opinion,  a  most  excellent  treatise  on  'the  Science  of  the  English  Lan 
guage.'  The  author  has  studiously  and  properly  excluded  from  hi's  book  the  technical 
ities,  jargon,  and  ambiguity  which  so  often  render  attempts  to  teach  grammar  unpleas 
ant,  if  not  impracticable.  The  inductive  plan  which  he  has  adopted,  and  of  which  he 
Is,  in  teaching  grammar,  the  originator,  is  admirably  adapted  to  t,le  great  purposes  of 
both  teaching  and  learning  the  important  science  of  our  language." — S.  JV.  Sweet,  Au 
thor  of  "Sweet's  Elocution.1'' 

P 


LOAN  DEPT 


